Astronomical Distances

Astronomical distances are very large and as a result, are usually measured using:
- Astronomical Units (AU)
- Light–years (ly)
- Parsecs (pc)

Astronomical Unit (AU)

The astronomical unit (AU) is defined as

The mean distance from the centre of the Earth to the centre of the Sun

As the Earth’s orbit around the Sun is elliptical it will be
- Slightly closer to the Sun in January (1.471 × 10¹¹ m)
- Slightly further away from the Sun in July (1.521 × 10¹¹ m)

Calculating the mean of these two values gives:

$\frac{(1.471 \times 10^{11}) + (1.521 \times 10^{11})}{2}$ = 1.496 × 10¹¹ m

Therefore, 1 astronomical unit (AU) = 1.496 × 10¹¹ m = 1.5 × 10¹¹ m
The astronomical unit is useful for studying distances on the scale of the solar system

How Far is an Astronomical Unit?

An astronomical unit (AU) is the mean distance between the Earth and the Sun

Light–year (ly)

A light-year is defined as:

The distance travelled by light in one year

This can be calculated using:

Distance = speed × time

Where:
- The speed of light is 3 × 10⁸ m s^–1
- 1 year = 60 × 60 × 24 × 365 = 3.15 × 10⁷ s
Hence, the distance travelled by light in one year = (3 × 10⁸) × (3.15 × 10⁷) = 9.46 × 10¹⁵ m
- Therefore, 1 light–year ≈ 9.5 × 10¹⁵ m

Parsec (pc)

Angles smaller than 1 degree can be measured in arcminutes or arcseconds
- 1 degree = 60 arcminutes
- 1 arcminute = 60 arcseconds
- Therefore, 1 degree = 60 × 60 = 3600 arcseconds
- 1 arcsecond = 1/3600 degree

The parsec is defined as

The distance at which the radius of the Earth's orbit (1 AU) around the Sun subtends at an angle of 1 arcsecond

How Far is a Parsec?

9-2-3-parsec

A parsec is defined using parallax angles

Given that 1 AU = 1.496 × 10¹¹ m, trigonometry can be used to express 1 parsec in metres:

$\tan θ = \frac{o p p}{a d j} = \frac{1 A U}{1 p c}$

$\tan (\frac{1}{3600}) = \frac{1 A U}{1 p c}$

1 pc = $\frac{1 A U}{\tan (\frac{1}{3600})} = \frac{1.496 \times 10^{11}}{\tan (\frac{1}{3600})}$ = 3.09 × 10¹⁶ m

Therefore, 1 parsec ≈ 3.1 × 10¹⁶ m
The parsec (1 pc = 3.1 × 10¹⁶ m) and the light-year (1 ly = 9.5 × 10¹⁵ m) are much greater in size than the astronomical unit (1 AU = 1.496 × 10¹¹ m)
This makes them useful for studying interstellar distances
- For example, on the scale of distances between the Earth and stars, or neighbouring galaxies

Worked example

The closest star to Earth is a triple–star system called Alpha Centauri, which is approximately 4.35 light-years from Earth.

Calculate the distance between the Earth and Alpha Centauri in:

(a)

astronomical units (AU)

(b)

parsecs (pc)

1 astronomical unit (AU) = 1.496 × 10¹¹ m.

Answer:

List the known quantities:

Distance to Alpha Centauri = 4.35 ly
1 AU = 1.496 × 10¹¹ m
1 light-year ≈ 9.5 × 10¹⁵ m (from data booklet)
1 parsec ≈ 3.1 × 10¹⁶ m (from data booklet)

Convert 4.35 light–years into metres:

distance = 4.35 ly = 4.35 × (9.5 × 10¹⁵)

distance = 4.13 × 10¹⁶ m

(a) Convert from metres into AU:

distance (AU) = $\frac{4.13 \times 10^{16}}{1.496 \times 10^{11}}$

distance (AU) = 2.8 × 10⁵ AU (2 s.f)

(b) Convert from metres into parsecs:

distance (pc) = $\frac{4.13 \times 10^{16}}{3.1 \times 10^{16}}$

distance (pc) = 1.3 pc (2 s.f)

Examiner Tip

You do not need to learn these conversion factors for astronomical distances, you just need to know how to use them. The following are given in the data booklet:

1 astronomical unit = 1.50 × 10¹¹ m

1 light year = 9.46 × 10¹⁵ m

1 parsec = 2.06 × 10⁵ AU = 3.08 × 10¹⁶ m = 3.26 ly

Astronomical Distances (AQA A Level Physics)

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