Hubble's law

Physics· Unit 29: Astronomy and cosmology· 15 min read

1. Hubble's Discovery and Law Statement★★☆☆☆⏱ 5 min

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.

📘 Definition

Hubble's Law

v = H_0 d

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.

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.

2. Physical Interpretation: Expanding Space★★★☆☆⏱ 5 min

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.

📐 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
1. 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.

✓ Quick check

Check your understanding:

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
Reveal answer
1 —
Correct! Uniform expansion of space produces Hubble's law for all observers, with no unique center of expansion.

3. Core Applications: Distance and Age Estimation★★★☆☆⏱ 5 min

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 $H_{0} = 2.2 \times 1 0^{- 18}$ s⁻¹. Estimate the age of the universe in years, to 1 significant figure. (1 year ≈ $3.15 \times 1 0^{7}$ 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
$T = 1 \times 10^{10} \text{ years (10 billion years)}$

4. Common Pitfalls

Wrong move:

Interpreting Hubble's law as galaxies moving through space away from a central Milky Way

Why:

This misrepresents cosmic expansion, which is expansion of space itself not motion through space

Correct move:

State that space expands uniformly, so all observers see the same Hubble relation with no unique center

Wrong move:

Rearranging Hubble's law incorrectly as $d = H_{0} / v$

Why:

This gives an inverse relationship between distance and velocity, contradicting Hubble's observation

Correct move:

Remember $v \propto d$ , so $v = H_{0} d$ , rearranged to $d = v / H_{0}$

Wrong move:

Forgetting to convert units when calculating age from $H_{0}$ in km s⁻¹ Mpc⁻¹

Why:

Units are inconsistent, leading to a wrong order of magnitude for the age of the universe

Correct move:

Convert $H_{0}$ to s⁻¹ first by converting Mpc to km, then calculate $1/ H_{0}$

Wrong move:

Claiming the Hubble constant is a true constant that never changes over time

Why:

$H_{0}$ only describes the current expansion rate, which has changed over the universe's history

Correct move:

Refer to $H_{0}$ as the present-day value of the universe's expansion rate

Wrong move:

Using Hubble's law for nearby galaxies in the Local Group

Why:

Local galaxies are gravitationally bound, so their motion is dominated by gravity not cosmic expansion

Correct move:

Only apply Hubble's law to distant galaxies not gravitationally bound to the Local Group

5. Quick Reference 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	$1 Mpc = 3.086 \times 1 0^{19} km$	Use to convert $H_{0}$ from km s⁻¹ Mpc⁻¹ to s⁻¹

6. Frequently Asked

Why is the Hubble constant not constant over time?

The Hubble constant $H_{0}$ refers to the current value of the universe's expansion rate. The expansion rate has changed over time due to gravity and dark energy, so it was different in the past.

Is Hubble redshift the same as Doppler redshift?

No. Hubble redshift (cosmological redshift) is caused by the expansion of space stretching light wavelength, while Doppler redshift is caused by objects moving through space. The two effects have different physical origins.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 22
Calculate galaxy distance from H0 and v
2023 · 13
Estimate universe age from given H0
2021 · 21
Explain significance of Hubble's law

Going deeper

externalESA: Expanding Universe OverviewVisual explanation of cosmic expansion

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.