Measuring astronomical distancesEdexcel A-Level Physics: Revision notes

As the Earth orbits the Sun, a nearby star appears to shift against the background of very distant stars. Measure its position, then again six months later, when the Earth is on the opposite side of its orbit. The parallax angle $p$ is half the total angular shift.

Geometrically, $p$ is the angle subtended at the star by the Earth-Sun distance, 1 astronomical unit (AU). For a very small angle $p \approx 1\,\text{AU}/d$.

One parsec (pc) is the distance at which a star has a parallax angle of one arcsecond ($1/3600$ of a degree). So

$d\,(\text{pc}) = \dfrac{1}{p\,(\text{arcseconds})}$

$1\,\text{pc} = 3.09 \times 10^{16}$ m, and an angle of $0.25$ arcseconds gives $d = 4.0$ pc.

A smaller parallax angle means a larger distance.

Parallax angles shrink with distance. Below about $0.01$ arcseconds the angle is too small to measure from the ground, so the method works only for nearby stars, up to a few hundred parsecs. A galaxy at $25$ Mpc would have $p = 1/(25 \times 10^{6}) = 4.0 \times 10^{-8}$ arcseconds, which is far too small to measure.

For greater distances another method is needed.

A standard candle is an object whose luminosity is known, such as a Type Ia supernova, which always reaches the same peak luminosity. If you measure the intensity $I$ received at the Earth, you can find its distance from $I = L/4\pi d^2$:

$d = \sqrt{\dfrac{L}{4\pi I}}$

Worked example. A supernova of $L = 2.0 \times 10^{36}$ W gives $I = 1.0 \times 10^{-14}$ W m⁻². Then $d = \sqrt{2.0 \times 10^{36} / (4\pi \times 1.0 \times 10^{-14})} = 4.0 \times 10^{24}$ m.

Standard candles reach much further than parallax, but depend on the luminosity being known and on negligible absorption by dust.

Parallax gives distances to nearby stars, and from these distances the luminosity of a star can be found, $L = 4\pi d^2 I$. For example, a star with $p = 0.050$ arcseconds is at $20$ pc ($6.2 \times 10^{17}$ m). With $I = 2.4 \times 10^{-10}$ W m⁻² its luminosity is $1.2 \times 10^{27}$ W.

That luminosity calibrates standard candles, which then give distances to galaxies far beyond the reach of parallax.