Cosmology: (i) Hubble constant H 0 in both km s –1 Mpc –1 and s –1 units

The idea

The Hubble constant H₀ links a galaxy's recession speed v to its distance d through v = H₀d, so H₀ is the gradient of a v–d graph. Its usual unit is km s⁻¹ Mpc⁻¹: a galaxy 1 Mpc away recedes at H₀ km s⁻¹. Because 1 Mpc = 3.09 × 10¹⁹ km, the km cancels when you divide, giving H₀ in s⁻¹. For example, 70 km s⁻¹ Mpc⁻¹ becomes 70 ÷ (3.09 × 10¹⁹) s⁻¹ ≈ 2.3 × 10⁻¹⁸ s⁻¹. The reciprocal 1/H₀ then has units of seconds and gives a rough age of the universe, about 4.4 × 10¹⁷ s or 1.4 × 10¹⁰ years. Always convert megaparsecs to kilometres before dividing, and keep the negative index on s⁻¹.

What earns the marks

• H₀ relates recession velocity to distance by v = H₀d, so it is the gradient of a velocity–distance graph.
• The standard unit km s⁻¹ Mpc⁻¹ means kilometres per second per megaparsec of distance.
• Converting: 1 Mpc = 3.09 × 10¹⁹ km, so km s⁻¹ Mpc⁻¹ divided by 3.09 × 10¹⁹ km gives s⁻¹.
• A value such as 70 km s⁻¹ Mpc⁻¹ converts to about 2.3 × 10⁻¹⁸ s⁻¹.
• The reciprocal 1/H₀ has units of seconds and estimates the age of the universe.

Common mistakes and how to put them right

• Multiplying by 3.09 × 10¹⁹ instead of dividing.

  Put it right: The conversion factor must cancel the km, so divide km s⁻¹ Mpc⁻¹ by 3.09 × 10¹⁹ km to obtain s⁻¹.

• Writing the unit as s¹ or s.

  Put it right: The quantity is a rate per second, so the correct unit is s⁻¹.

• Treating Mpc as a time unit.

  Put it right: A megaparsec is a distance, 3.09 × 10¹⁹ km, not a duration.

Exam tips

• Write the conversion factor 1 Mpc = 3.09 × 10¹⁹ km explicitly before substituting numbers.
• Check the unit by cancelling: km s⁻¹ Mpc⁻¹ × (Mpc / 3.09 × 10¹⁹ km) leaves s⁻¹.
• If asked for the age of the universe, quote 1/H₀ in seconds and convert to years by dividing by 3.15 × 10⁷ s year⁻¹.