Measuring astronomical distancesEdexcel International A Level Physics: Revision notes

Distances in space are so large that the metre is inconvenient.

• The astronomical unit (AU) is the mean distance from the Earth to the Sun, $1.50\times10^{11}$ m.
• A light-year (ly) is the distance light travels in one year in a vacuum, about $9.46\times10^{15}$ m.
• A parsec (pc) is the distance at which a star would have a parallax angle of one arcsecond, $3.09\times10^{16}$ m, which is about 3.26 ly.

An arcsecond is $\dfrac{1}{3600}$ of a degree.

As the Earth orbits the Sun, a nearby star appears to shift against the much more distant background stars. This apparent shift is called parallax.

1. Measure the position of the star against distant stars.
2. Measure it again six months later, when the Earth is on the opposite side of its orbit.
3. Half of the total angular shift is the parallax angle $p$. It is the angle subtended at the star by a distance of 1 AU.

For a small angle, $\tan p\approx p$, so the distance in AU is $d=\dfrac{1}{p}$ with $p$ in radians. By the definition of the parsec:

$d\text{ (pc)}=\frac{1}{p\text{ (arcsecond)}}$

The nearer the star, the larger its parallax angle.

The parallax angle is inversely proportional to the distance. For distant stars it becomes smaller than the smallest angle a telescope can resolve, and atmospheric blurring makes small angles harder to measure from the ground.

So parallax works only for relatively nearby stars. For more distant objects, astronomers use standard candles.

The luminosity $L$ of a star is the total power it radiates, in watts.

The intensity $I$ is the power received per unit area at the observer, in W m⁻². Light spreads over a sphere of area $4\pi d^2$, so

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

Intensity obeys the inverse square law: doubling the distance reduces the intensity to one quarter.

A standard candle is an object of known luminosity. Examples are type Ia supernovae, which are extremely luminous and all reach nearly the same peak luminosity.

To find the distance to a standard candle:

1. Identify an object whose luminosity $L$ is known.
2. Measure the intensity $I$ received at Earth.
3. Rearrange: $d=\sqrt{\dfrac{L}{4\pi I}}$.

Worked example. A supernova with $L=2.0\times10^{36}$ W has $I=1.6\times10^{-13}$ W m⁻² at Earth.

$d=\sqrt{\dfrac{2.0\times10^{36}}{4\pi\times1.6\times10^{-13}}}=1.0\times10^{24}$ m

The luminosity of a standard candle is calibrated using nearby examples whose distances are found by parallax, so errors in nearby distances pass on to distant ones.

Other sources of uncertainty:

• dust absorbs some light, so $I$ is too low and $d$ is overestimated
• the assumption that all standard candles of one type have identical luminosity may not be exact

Standard candles reach much further than parallax but are less precise.