5.12.1 Units for 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) than it is in July (1.521 × 10¹¹ m)
• Calculating the mean of these two values gives:

= 1.496 × 10¹¹ m

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

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 x 60 = 3600 arcseconds
  • 1 arcsecond = 1/3600 degree

• The parsec is defined as

A unit of distance that gives a parallax angle of 1 second of an arc (of a degree), using the radius of the Earth’s orbit (1 AU) as the baseline of a right–angled triangle

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

1 pc =  = 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 when studying interstellar distances
  • For example, on the scale of distances between the Earth and stars, or neighbouring galaxies

Part (a)

Step 1: List the known quantities

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

Step 2: Convert 4.35 light–years into metres

• • 4.35 ly = 4.35 × (9.5 × 10¹⁵) = 4.13 × 10¹⁶ m

Step 3: Convert from metres into AU

• • 4.13 × 10¹⁶ m = = 2.8 × 10⁵ AU (to 2 s.f)

Part (b)

Step 1: List the known quantities

• • 1 parsec ≈ 3.1 × 10¹⁶ m (from data booklet)
  • 4.35 ly = 4.13 × 10¹⁶ m (from part a)

Step 2: Convert from metres into parsecs

• • 4.13 × 10¹⁶ m =  = 1.3 pc (to 2 s.f)