# Units for Astronomical Distances ## Why Special Units? - Distances in space are enormous compared to everyday measurements - Using metres would result in impractically large numbers (like $1.5 \times 10^{11}$ m for Earth-Sun distance) - Three specialized units make astronomical calculations more manageable: - **Astronomical Unit (AU)** - for solar system scales - **Light-year (ly)** - for stellar distances - **Parsec (pc)** - for interstellar and galactic distances --- ## Astronomical Unit (AU)

The mean distance between the centre of the Earth and the centre of the Sun

### Why "mean" distance? - Earth's orbit is elliptical, not circular - Distance varies throughout the year: - Closest approach (perihelion): $1.471 \times 10^{11}$ m - Farthest point (aphelion): $1.521 \times 10^{11}$ m ### Calculating 1 AU $$\text{Mean distance} = \frac{1.471 \times 10^{11} + 1.521 \times 10^{11}}{2} = 1.496 \times 10^{11} \text{ m}$$ $$\boxed{1 \text{ AU} \approx 1.5 \times 10^{11} \text{ m}}$$ ### When to use AU - Ideal for describing distances within our solar system - Example: Mars orbits at roughly 1.5 AU from the Sun - Example: Neptune orbits at approximately 30 AU from the Sun --- ## Light-year (ly)

The distance travelled by light in a vacuum in one year

### Calculating 1 light-year Use the fundamental equation: $$\text{distance} = \text{speed} \times \text{time}$$ **Step 1:** Identify the values - Speed of light: $c = 3.00 \times 10^8$ m s$^{-1}$ - Time in one year: $60 \times 60 \times 24 \times 365 = 3.15 \times 10^7$ s **Step 2:** Calculate the distance $$\text{distance} = (3.00 \times 10^8) \times (3.15 \times 10^7) = 9.45 \times 10^{15} \text{ m}$$ $$\boxed{1 \text{ ly} \approx 9.5 \times 10^{15} \text{ m}}$$ ### When to use light-years - Useful for distances to nearby stars - Example: Proxima Centauri (nearest star to our Sun) is 4.2 ly away - Commonly used in popular science and astronomy outreach --- ## Parsec (pc) ### Understanding arcseconds Before defining the parsec, we need to understand angular measurements: | Unit | Conversion | |------|------------| | 1 degree | 60 arcminutes (60') | | 1 arcminute | 60 arcseconds (60") | | 1 degree | 3600 arcseconds (3600") | Therefore: $1 \text{ arcsecond} = \frac{1}{3600}$ degree

A parsec is the distance at which 1 AU subtends an angle of 1 arcsecond.

### The geometry of a parsec - Imagine a right-angled triangle: - Base = 1 AU (Earth's orbital radius) - Angle at apex = 1 arcsecond - Height = distance to star = 1 parsec ### Calculating 1 parsec using trigonometry Starting with the tangent ratio: $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1 \text{ AU}}{1 \text{ pc}}$$ For angle $\theta = 1$ arcsecond $= \frac{1}{3600}$ degree: $$\tan\left(\frac{1}{3600}\right) = \frac{1 \text{ AU}}{1 \text{ pc}}$$ Rearranging: $$1 \text{ pc} = \frac{1 \text{ AU}}{\tan\left(\frac{1}{3600}\right)}$$ $$1 \text{ pc} = \frac{1.496 \times 10^{11}}{\tan\left(\frac{1}{3600}\right)} = 3.09 \times 10^{16} \text{ m}$$ $$\boxed{1 \text{ pc} \approx 3.1 \times 10^{16} \text{ m}}$$ ### When to use parsecs - Standard unit for professional astronomy - Used for stellar and galactic distance measurements - Links directly to stellar parallax measurements - Example: The centre of our Milky Way galaxy is approximately 8,000 pc away --- ## Comparing the Three Units | Unit | Value in metres | Best used for | |------|----------------|---------------| | 1 AU | $1.5 \times 10^{11}$ m | Solar system distances | | 1 ly | $9.5 \times 10^{15}$ m | Stellar distances | | 1 pc | $3.1 \times 10^{16}$ m | Interstellar & galactic distances | **Key observation:** Both ly and pc are much larger than AU - This makes them suitable for the vast distances between stars - AU would be too small and cumbersome for these scales ### Visualizing the magnitude differences The chart below shows the relative sizes of these units in metres (displayed on a logarithmic-like scale due to the vast differences):

- Light-years are approximately **63,000 times** larger than astronomical units - Parsecs are approximately **207,000 times** larger than astronomical units - Parsecs are roughly **3.3 times** larger than light-years --- ## Worked Example: Converting Between Units **Question:** Betelgeuse, a red supergiant star in the constellation Orion, is approximately 550 light-years from Earth. Calculate this distance in: a) Astronomical units b) Parsecs **Answer:** ### Part (a): Converting to AU **Step 1:** Write down known values - Distance = 550 ly - 1 ly $\approx 9.5 \times 10^{15}$ m (from data booklet) - 1 AU $= 1.496 \times 10^{11}$ m **Step 2:** Convert light-years to metres $$550 \text{ ly} = 550 \times (9.5 \times 10^{15}) = 5.225 \times 10^{18} \text{ m}$$ **Step 3:** Convert metres to AU $$\text{Distance in AU} = \frac{5.225 \times 10^{18}}{1.496 \times 10^{11}} = 3.5 \times 10^{7} \text{ AU}$$ **Answer:** Betelgeuse is approximately **35 million AU** from Earth ### Part (b): Converting to parsecs **Step 1:** Use the distance in metres from part (a) - Distance $= 5.225 \times 10^{18}$ m **Step 2:** Write down the parsec conversion - 1 pc $\approx 3.1 \times 10^{16}$ m (from data booklet) **Step 3:** Convert metres to parsecs $$\text{Distance in pc} = \frac{5.225 \times 10^{18}}{3.1 \times 10^{16}} = 1.7 \times 10^{2} \text{ pc} = 170 \text{ pc}$$ **Answer:** Betelgeuse is approximately **170 parsecs** from Earth ---

The data booklet provides: 1 ly ≈ 9.5 × 10¹⁵ m and 1 pc ≈ 3.1 × 10¹⁶ m. The AU is not listed, so it's worth memorizing: 1 AU ≈ 1.5 × 10¹¹ m.

When converting between units, always go through metres as an intermediate step unless you have direct conversion factors memorized