The earth goes around the sun once a year, so where it observes from moves by the width of the orbit. Seen from two places apart, a nearby object appears in a different direction. This is the same as holding up a finger and closing one eye and then the other.
Against the far background of stars, a nearby star traces a small ellipse over a year and returns to where it began. The nearer the star, the larger this apparent swing.
Half of that swing, measured as an angle, is called the annual parallax and is written . The parallax is the angle the radius of the orbit of the earth subtends as seen from the star.
The background stars do not move because they are far enough away that the width of the orbit counts for nothing. What moves and what does not, in the same field of view, is itself the measurement.
A long thin triangle is formed by the sun, the earth and the star. Since the angle is very small, the distance is inversely proportional to the parallax.
The distance at which the parallax comes to second of arc is taken as the unit and called parsec. With in seconds of arc and in parsecs, this is simply .
A parallax of seconds of arc means parsecs, and seconds of arc means parsecs. Halve the angle and the distance doubles.
One parsec is about light years. The unit was made for this measurement, which is why the formula comes out with nothing left over.
One second of arc is of a degree. The full moon is about degrees across, which is seconds of arc, so one second of arc is about of the moon.
Even the nearest star, in the Centaurus constellation, has a parallax of only seconds of arc. Every star has a parallax smaller than one second of arc, without exception.
This is why the measurement waited so long. Copernicus argued that the earth goes around the sun, but no swing of the stars could be found, which was held against him. The angle was simply too small to reach.
It was first measured in 1838. Instruments finally reached below one second of arc, and the motion of the earth showed itself in the sky at the same time.
The light that arrives falls off as the square of the distance. Turn this around: multiply the apparent brightness by the square of the distance and the amount actually emitted, the luminosity , follows.
Two stars can look equally bright while being nothing alike. If one is times further away, the light it emits is times greater.
This is why brightness in the sky says nothing by itself. Distance has to come first, and only then does the star itself come into view.
The vertical axis of the HR diagram is this luminosity. Every star plotted there had its distance measured first.
Parallax runs out. The further the star, the smaller the angle, until it disappears into the error of the instrument. What lies beyond has to be reached another way.
The HR diagram is what carries it. Split the light of a star and its surface temperature follows, which fixes a position along the main sequence, which gives the luminosity.
With the luminosity in hand, compare it against the apparent brightness and the distance follows from the inverse square. This is called the spectroscopic parallax, though no angle is measured in it.
Parallax is the first rung and the HR diagram the second, each calibrated on the one below. Measuring the far universe is built as a ladder of such rungs, and this is where it starts.