# Hubble's law

> Physics · CIE A-Level
> Source: https://www.owlsprep.com/study/cie-9702-u29-hubble-s-law/

This subtopic covers Edwin Hubble's discovery of the expanding universe, the mathematical statement of Hubble's law, and its core applications including calculating cosmic distances and estimating the age of the universe for CIE A-Level Physics.

**Prerequisites:** [Redshift and Doppler effect](https://www.owlsprep.com/study/cie-9702-u29-redshift-doppler-effect/); [Cosmic distance measurement](https://www.owlsprep.com/study/cie-9702-u29-cosmic-distance-measurement/)

## Learning objectives

- State Hubble's law and explain its physical meaning
- Calculate recessional velocity or distance using Hubble's law
- Estimate the age of the universe from the Hubble constant
- Link Hubble's law to the theory of an expanding universe

## Hubble's Discovery and Law Statement

In the 1920s, Edwin Hubble measured distances to nearby galaxies using Cepheid variable stars, and compared these distances to each galaxy's measured redshift. He found a clear linear relationship between a galaxy's distance from Earth and its recessional velocity.

**Hubble's Law** — Hubble's law states that the recessional velocity $v$ of a distant galaxy is directly proportional to its distance $d$ from the observer. The constant of proportionality $H_0$ is the Hubble constant, the current rate of expansion of the universe.

*Notation:* v = H_0 d

*Example:* A galaxy 10 Mpc away recedes at ~700 km/s if $H_0 = 70$ km s⁻¹ Mpc⁻¹

**Worked example:** A galaxy has a measured recessional velocity of 1500 km s⁻¹. Given $H_0 = 75$ km s⁻¹ Mpc⁻¹, calculate the distance to the galaxy.

1. Rearrange Hubble's law to isolate distance $d$:
2. $$d = \frac{v}{H_0}$$
3. Substitute the given values:
4. $$d = \frac{1500 \text{ km s}^{-1}}{75 \text{ km s}^{-1} \text{ Mpc}^{-1}} = 20 \text{ Mpc}$$

> **Exam tip:** Always check units for $H_0$: CIE commonly gives $H_0$ in both km s⁻¹ Mpc⁻¹ and s⁻¹, ensure your answer matches the required unit for distance.

## Physical Interpretation: Expanding Space

Hubble's law is often misinterpreted as meaning galaxies move *through* space away from a central point at Earth. In fact, the law is a consequence of space itself expanding uniformly. Every observer in any galaxy will observe the same linear relationship between distance and recessional velocity, so the universe has no unique center of expansion.

> **info**
>
> Cosmological redshift (the redshift measured for distant galaxies) is caused by expansion of space stretching the wavelength of light as it travels to Earth, not by the Doppler effect of motion through space.

**Worked example:** Explain why all observers in any galaxy observe Hubble's law for a uniformly expanding universe.

1. Model the universe as the surface of an expanding balloon with dots marking galaxies:
2. 1. As the balloon inflates, any two dots move further apart, and the rate of increase of distance between them is proportional to their current separation.
3. 2. An observer on any dot will see all other dots moving away, with speed proportional to distance, exactly matching Hubble's law.
4. No dot is at the 'center' of expansion, because expansion is of the entire surface (space) itself.

**Check your understanding**

Check your understanding:

1. Which of the following is the correct interpretation of Hubble's law?

   - All galaxies move away from the center of the universe through space
   - Space expands uniformly, so recessional speed increases with distance from any observer
   - Hubble's law only applies to observations made from the Milky Way

   *Answer:* Space expands uniformly, so recessional speed increases with distance from any observer

   *Why:* Correct! Uniform expansion of space produces Hubble's law for all observers, with no unique center of expansion.

## Core Applications: Distance and Age Estimation

Hubble's law has two key applications tested in CIE A-Level: calculating distances to far galaxies (once recessional velocity is measured from redshift), and estimating the approximate age of the universe. The approximate age, called the Hubble time, is given by $1/H_0$, assuming the expansion rate has remained constant over time.

**Worked example:** The Hubble constant is $H_0 = 2.2 \times 10^{-18}$ s⁻¹. Estimate the age of the universe in years, to 1 significant figure. (1 year ≈ $3.15 \times 10^7$ s)

1. The approximate age of the universe $T$ equals the Hubble time $1/H_0$:
2. $$T = \frac{1}{H_0} = \frac{1}{2.2 \times 10^{-18} \text{ s}^{-1}} = 4.55 \times 10^{17} \text{ s}$$
3. Convert seconds to years by dividing by the number of seconds in a year:
4. $$T = \frac{4.55 \times 10^{17}}{3.15 \times 10^7} \approx 1.44 \times 10^{10} \text{ years}$$
5. Round to 1 significant figure as required:
6. $$T = 1 \times 10^{10} \text{ years (10 billion years)}$$

**Exam command terms**

CIE uses the following common command terms for this topic:

- **Estimate** — The approximation $T = 1/H_0$ is expected, no more complex calculation is required *(Estimate the age of the universe from the given $H_0$)*

- **Explain the significance** — You must link Hubble's observation to the expanding universe theory, not just state the law *(Explain the significance of Hubble's law for cosmology)*

## Common pitfalls

- **Wrong:** Interpreting Hubble's law as galaxies moving through space away from a central Milky Way
  - Why it fails: This misrepresents cosmic expansion, which is expansion of space itself not motion through space
  - Correct: State that space expands uniformly, so all observers see the same Hubble relation with no unique center
- **Wrong:** Rearranging Hubble's law incorrectly as $d = H_0/v$
  - Why it fails: This gives an inverse relationship between distance and velocity, contradicting Hubble's observation
  - Correct: Remember $v \propto d$, so $v = H_0 d$, rearranged to $d = v/H_0$
- **Wrong:** Forgetting to convert units when calculating age from $H_0$ in km s⁻¹ Mpc⁻¹
  - Why it fails: Units are inconsistent, leading to a wrong order of magnitude for the age of the universe
  - Correct: Convert $H_0$ to s⁻¹ first by converting Mpc to km, then calculate $1/H_0$
- **Wrong:** Claiming the Hubble constant is a true constant that never changes over time
  - Why it fails: $H_0$ only describes the current expansion rate, which has changed over the universe's history
  - Correct: Refer to $H_0$ as the present-day value of the universe's expansion rate
- **Wrong:** Using Hubble's law for nearby galaxies in the Local Group
  - Why it fails: Local galaxies are gravitationally bound, so their motion is dominated by gravity not cosmic expansion
  - Correct: Only apply Hubble's law to distant galaxies not gravitationally bound to the Local Group

## Cheatsheet

| Concept | Formula | Key Notes |
| --- | --- | --- |
| Hubble's Law | $v = H_0 d$ | $v$ = recessional velocity, $d$ = distance, $H_0$ = current expansion rate |
| Calculate distance | $d = v/H_0$ | Match units of $H_0$ to required distance unit |
| Estimate universe age | $T = 1/H_0$ | Convert $H_0$ to s⁻¹ to get age in seconds |
| Interpretation | Uniform expansion of space | *No* center of expansion, all observers see same law |
| Common unit conversion | $1 \text{ Mpc} = 3.086 \times 10^{19} \text{ km}$ | Use to convert $H_0$ from km s⁻¹ Mpc⁻¹ to s⁻¹ |

## What's next

Hubble's law is the foundational observational evidence for the expanding universe, and underpins all modern Big Bang cosmology. After mastering this topic, you will build on this knowledge to explore the Big Bang model itself, the cosmic microwave background (the residual heat from the early universe), and current theories about the fate of the universe. Hubble's law also connects to modern open questions like the Hubble tension, where different measurements of $H_0$ give conflicting results, driving current research in cosmology.

- [Big Bang cosmology](https://www.owlsprep.com/study/cie-9702-u29-big-bang-cosmology/)
- [Practical skills (A2)](https://www.owlsprep.com/study/cie-9702-u30-overview/)
- [Advanced Experiment Planning](https://www.owlsprep.com/study/cie-9702-u30-advanced-experiment-planning/)

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