9.3.2 Hubble's Law

Hubble's Law Overview:

• Hubble's Law states that the recessional velocity of a galaxy (how fast it is moving away from us) is directly proportional to its distance from Earth. This implies that the universe is expanding from a single, common starting point.
• Mathematically, Hubble's Law is expressed as:

$$v = H \cdot d$$

Where:

• $v$ = recessional velocity (km/s),
• $H$ = Hubble constant (approximately 65 km/s/Mpc),
• $d$ = distance of the galaxy from Earth (in megaparsecs, Mpc).

Estimating the Age of the Universe:

1. Start with the formula for Hubble's Law: $v = H \cdot d$.
2. Rearrange to find time, assuming that $\text{time} = \frac{\text{distance}}{\text{velocity}}$:

$$t = \frac{1}{H}$$

3. Convert units of $H$ (from km/s/Mpc) to SI units for consistency in calculations:

• Multiply $H$ by 1000 to convert km to m.
• Divide by $1$ Mpc to convert megaparsecs to metres.

4. Using these values, calculate:

$$t = \frac{1}{2.1 \times 10^{-18}} \approx :highlight[4.74 \times 10^{17}] \text{ seconds}$$

5. Convert seconds to years:

$$4.74 \times 10^{17} \div 3.16 \times 10^7 \approx :highlight[15 billion years]$$

6. This provides an estimate for the age of the universe.