Year 12 Physics Module 8 ⏱ ~20 min 5 MC · 2 Short Answer Lesson 3 of 17

Hubble's Law and the Expanding Universe

You'll be able to calculate redshift from spectral data.

In 1929, Edwin Hubble at Mount Wilson Observatory published redshift and distance measurements for 46 galaxies, finding that recession velocity and distance are proportional. His original Hubble constant was H₀ ≈ 500 km/s/Mpc, far larger than the modern value of 67.4 km/s/Mpc (Planck 2018), but the linear relationship was correct. At the Hubble radius d = c/H₀ ≈ 4,400 Mpc, recession speed equals the speed of light. Hubble's 1929 velocity-distance graph is one of the most reproduced plots in the history of physics.

Today's hook: In 1929, Edwin Hubble at Mount Wilson Observatory measured the redshifts and distances of 46 galaxies, then plotted velocity against distance. He found a straight line with slope H₀ ≈ 500 km/s/Mpc: a galaxy 1 Mpc away recedes at 500 km/s; one 2 Mpc away at 1,000 km/s. The modern value is H₀ = 67.4 km/s/Mpc (Planck 2018). What does this linear relationship tell us about the geometry of the expanding universe, and why does it imply no centre?
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Orient and predict

Set up the vocabulary and units for redshift and distance, then predict what one shifted spectral line tells you.

Warm up first

Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.

Worksheets

Practise this lesson

Four printable worksheets that build from the foundations up to exam-style questions, start at whatever level suits you.

Build Foundations & guided practice Apply Application & data response Master Mastery tasks Exam Exam-style questions
Before you read, predict

A galaxy's hydrogen emission line is observed at 656.5 nm instead of the laboratory value 656.3 nm.

Before reading on, answer:

  1. Is this galaxy moving toward us or away from us?
  2. Calculate its recession velocity (use $c = 3.00\times10^8$ m/s).
  3. If Hubble's constant is 70 km/s/Mpc, estimate its distance.

Warm-up: Hubble's law states that the recession velocity of a galaxy is proportional to its:

Learning Intentions
goals

Know, Hubble's Law

  • $v = H_0 d$
  • $H_0 \approx 70$ km/s/Mpc
  • Recession velocity proportional to distance

Understand, Scale Factor and Redshift

  • $1 + z = a_{\mathrm{now}}/a_{\mathrm{then}}$
  • Lookback time vs distance
  • Cosmological redshift

Can Do, Solve Cosmological Problems

  • Calculate redshift from spectral data
  • Estimate distances from Hubble's law
  • Convert between redshift and lookback time
Scan these before reading
vocab
Hubble constant ($H_0$)The rate of expansion of the universe, $\approx 70$ km/s/Mpc. Its inverse gives the Hubble time, a rough estimate of the universe's age.
Megaparsec (Mpc)$3.086\times10^{19}$ km or $\approx 3.26$ million light-years; the standard unit for cosmological distances.
Redshift ($z$)The fractional increase in wavelength of light from a receding source: $z = (\lambda_{\mathrm{obs}} - \lambda_{\mathrm{rest}})/\lambda_{\mathrm{rest}}$.
Scale factor ($a$)A dimensionless measure of the relative size of the universe; $a = 1$ today by convention. In the past, $a < 1$.
Lookback timeThe time light has been travelling to reach us from a distant object. For $z \ll 1$: $t_{lookback} \approx d/c$.
Cross-lesson links: L02 surveyed the three pillars of Big Bang evidence. L03 focuses on Hubble's law, the direct observational evidence that the universe is expanding. The Doppler effect (M7 L01) and spectroscopy (M7 L09) are the two M7 tools that make Hubble's measurement possible; L03 connects cosmology directly to your wave physics.

Core Content

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Read the Hubble graph

Read velocity against distance off the Hubble diagram, identify the gradient as $H_0$ and pin down its units.

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Hubble's Law
+5 XP

Velocity proportional to distance

Observe the spectrum of any distant galaxy through a telescope: the hydrogen absorption lines appear at wavelengths slightly longer than the same lines measured in a laboratory. Measure this redshift for many galaxies and compare it to their independently determined distances, and a strikingly simple pattern emerges. In 1929, Edwin Hubble at Mount Wilson Observatory published exactly this result for 46 galaxies, finding that recession velocity $v$ and distance $d$ are proportional:

$$v = H_0 d$$

The constant of proportionality, $H_0$, is the Hubble constant. Modern measurements give $H_0 \approx 70$ km/s/Mpc, though there is tension between different measurement methods (the "Hubble tension"). For HSC purposes, use $H_0 = 70$ km/s/Mpc unless otherwise stated.

The units are important: velocity in km/s, distance in megaparsecs (Mpc), so $H_0$ has units km/s/Mpc. One Mpc $= 3.086\times10^{19}$ km $\approx 3.26$ million light-years.

What Hubble's law means: If galaxy A is twice as far as galaxy B, it recedes twice as fast. This is exactly what you would expect if space itself is expanding uniformly, like dots on an inflating balloon, all moving apart with speed proportional to their separation.

Hubble diagram: galaxies’ recession velocities against distance scatter about a straight best-fit line through the origin, whose gradient is H₀ ≈ 70 km/s/Mpc.70001400021000280000100200300400500slope = H₀≈ 70 km/s/MpcDistance (Mpc)recession velocity (km/s)Points scatter either side of thebest-fit line through the origin: v = H₀d
Figure 1, Hubble diagram: recession velocity vs distance for galaxies. The gradient of the best-fit line is the Hubble constant $H_0 \approx 70$ km/s/Mpc.
Text description
  • A graph of recession velocity (0 to 35 000 km/s) against distance (0 to 500 Mpc). Each dot is a galaxy.
  • The dots scatter either side of a straight best-fit line through the origin, v = H₀d.
  • The gradient of the line is the Hubble constant, H₀ ≈ 70 km/s/Mpc: for example about 28 000 km/s at 400 Mpc.
Hubble's Law and Related Equations

$v = H_0 d$, Hubble's law ($H_0 \approx 70$ km/s/Mpc)

$z = v/c$, Redshift (low-$z$ approximation, $z \ll 1$)

$d = cz/H_0$, Distance from redshift (low-$z$)

$t_{H} = 1/H_0 \approx 14$ Gyr, Hubble time (approximate age of universe)

Stop and check

A galaxy has redshift $z = 0.05$. Estimate its recession velocity and its distance using Hubble's law ($H_0 = 70$ km/s/Mpc). Express the distance in Mpc and in light-years.

Hubble's law states $v = H_0 d$ ($H_0 \approx 70$ km/s/Mpc), so recession velocity is proportional to distance, the hallmark of uniform expansion of space. The Hubble diagram (plot of $v$ vs $d$) has gradient $H_0$; the Hubble time $t_H = 1/H_0 \approx 14$ Gyr gives a rough age of the universe.

Pause, copy the highlighted law and its terms into your book before moving on.

A galaxy at 200 Mpc has a recession velocity of approximately:

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Calculate distance from a redshift

Chain the steps in order: measured wavelength shift to $z$, then to $v$, then to distance and lookback time.

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Worked Example: Hubble's Law Calculations
+5 XP

From spectra to cosmic distances

We just saw that Hubble's law $v = H_0 d$ describes the proportional relationship between recession velocity and distance. That raises a question: how do we actually chain together the formulas to go from an observed spectral shift to a cosmic distance? This card answers it → step by step: find $z$, then $v = cz$, then $d = v/H_0$, then lookback time $\approx d/c$.

Problem

A distant galaxy shows the H$\alpha$ emission line at 714 nm. The laboratory wavelength is 656 nm.

  1. Calculate the redshift $z$.
  2. Estimate the recession velocity.
  3. Estimate the distance using Hubble's law ($H_0 = 70$ km/s/Mpc).
  4. Calculate the approximate lookback time.
  5. Compare this to the age of the universe (~13.8 Gyr).
Solution
  1. Redshift. $$z = \frac{714 - 656}{656} = \frac{58}{656} = \mathbf{0.088}$$
  2. Recession velocity. $$v = cz = (3.00\times10^5\ \text{km/s})(0.088) = \mathbf{2.64\times10^4\ \text{km/s}}$$
  3. Distance. $$d = \frac{v}{H_0} = \frac{26400}{70} = \mathbf{377\ \text{Mpc}}$$ $$377\ \text{Mpc} \times 3.26 = \mathbf{1230\ \text{Mly}} \approx 1.23\ \text{Gly}$$
  4. Lookback time. $$t \approx \frac{d}{c} = \frac{1230}{1000} \approx \mathbf{1.23\ \text{Gyr}}$$
  5. Comparison. Light left the galaxy when the universe was about $13.8 - 1.2 = 12.6$ Gyr old, about 90% of its current age.
Stop and check

A galaxy at 150 Mpc has what recession velocity? If its H$\beta$ line (rest 486 nm) is observed, what wavelength would you measure?

To find cosmic distance from a spectral shift: (1) $z = (\lambda_{\mathrm{obs}}-\lambda_{\mathrm{rest}})/\lambda_{\mathrm{rest}}$, (2) $v = cz$, (3) $d = v/H_0$, (4) lookback time $\approx d/c$. If $\lambda_{\mathrm{obs}} > \lambda_{\mathrm{rest}}$ the source is receding (redshift); 1 Mpc $\approx 3.26 \times 10^6$ ly.

Add the highlighted procedure to your notes before the check below.

An observed wavelength longer than the rest wavelength indicates that the galaxy is receding from us.

A galaxy twice as far away recedes at half the velocity, according to Hubble's law.

The Hubble time $1/H_0$ gives a rough estimate of the age of the universe.

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Link redshift to the scale factor

Turn an observed redshift into the size of the universe when the light left, and into a lookback time.

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Redshift and the Scale Factor
+5 XP

Connecting observation to cosmic history

We just saw how to calculate recession velocity and distance from a redshift using Hubble's law. That raises a question: what does the redshift $z$ actually tell us about the physical size of the universe when the light was emitted? This card answers it → via the scale factor $a = 1/(1+z)$, which directly links an observed redshift to the universe's relative size at that moment in history.

The scale factor $a(t)$ describes the relative size of the universe at time $t$. By convention, $a(t_{now}) = 1$. In the past, $a < 1$; in the future, $a > 1$.

Redshift is directly related to the scale factor:

$$1 + z = \dfrac{a_{\mathrm{now}}}{a_{\mathrm{then}}} = \dfrac{1}{a_{\mathrm{then}}}$$

So $a_{\mathrm{then}} = 1/(1+z)$. At $z = 1$, the universe was half its present size. At $z = 9$, it was one-tenth its present size.

Lookback time is the time light has been travelling to reach us. For nearby galaxies ($z \ll 1$):

$$t_{lookback} \approx \dfrac{d}{c} = \dfrac{z}{H_0}$$

For more distant objects, the relationship is non-linear because the expansion rate has changed over time. The Hubble time $t_H = 1/H_0 \approx 14$ Gyr gives a rough estimate of the age of the universe.

The universe’s relative size a against redshift z: a = 0.1 at z = 9, 0.2 at z = 4, 0.5 at z = 1, 0.83 at z = 0.2, and 1 today; 1 + z = 1/a.z=9a=0.1z=4a=0.2z=1a=0.5z=0.2a=0.83z=0a=1Big Banguniverse expandstoday1 + z = 1/a: higher z means a smaller,younger universe
Figure 2, The universe has expanded over cosmic time. At redshift $z$, the universe was a fraction $a = 1/(1+z)$ of its current size. Higher $z$ = smaller, younger universe.
Text description
  • Five circles show the size of the universe, relative to today, at five redshifts. Each radius is proportional to the scale factor a.
  • z = 9: a = 0.1. z = 4: a = 0.2. z = 1: a = 0.5. z = 0.2: a = 0.83. Today, z = 0 and a = 1.
  • 1 + z = 1/a, so a higher redshift means a smaller, younger universe.
Stop and check

A quasar has $z = 6.0$. What was the scale factor of the universe when its light was emitted? If $H_0 = 70$ km/s/Mpc, what is the approximate lookback time (using $t \approx z/H_0$ for a rough estimate)?

The scale factor $a = 1/(1+z)$ gives the universe's size relative to today when observed light was emitted: at $z = 1$, the universe was half its present size; at $z = 9$, one-tenth. The Hubble time $t_H = 1/H_0 \approx 14$ Gyr is a rough upper estimate of the universe's age (actual age ~13.8 Gyr, slightly less because gravity slowed early expansion).

Pause, write the highlighted formula and example values into your book before the check below.

A galaxy has $z = 0.10$. Using $v = cz$ with $c = 3.00 \times 10^5$ km/s, its recession velocity is _____ km/s.

5

Apply the law, watching units

Check the unit traps and the low-redshift condition, then run the calculations yourself.

HSC Tip, Hubble's Law Calculations

When using Hubble's law, watch your units carefully. $H_0 = 70$ km/s/Mpc means if $d$ is in Mpc, $v$ comes out in km/s. To convert Mpc to light-years, multiply by 3.26 million. A common trap: using $v = cz$ for large $z$ (>0.1). For $z = 0.5$, $v = 0.5c$ is only an approximation, the actual recession velocity depends on the cosmological model. In HSC, unless told otherwise, $v = cz$ is acceptable. Another trap: confusing redshifted (longer) with blueshifted (shorter) wavelengths. Redshift means $\lambda_{\mathrm{obs}} > \lambda_{\mathrm{rest}}$.

Activity 1, Hubble's Law Calculations
ApplyBand 4

Practice using $v = H_0 d$ and the redshift formula

  1. A galaxy has a recession velocity of 4200 km/s. Use Hubble's law ($H_0 = 70$ km/s/Mpc) to find its distance in Mpc and in light-years.
  2. A spectral line at rest wavelength 500 nm is observed at 550 nm. Calculate the redshift, recession velocity, and distance to this galaxy.
  3. A galaxy is at 300 Mpc. Calculate: (a) its recession velocity, (b) its redshift $z$, (c) the observed wavelength of H$\alpha$ (rest 656 nm).
  4. Explain why the Hubble time $1/H_0$ overestimates the true age of the universe.
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Reason about expansion and the model's limits

Argue why proportionality means expanding space with no centre, and where $v = cz$ stops working.

Activity 2, Reasoning About Expansion
UnderstandBand 5

Explain why Hubble's law implies expanding space

  1. Explain why Hubble's law ($v \propto d$) is evidence that space itself is expanding, rather than galaxies simply flying through a fixed space.
  2. If you were an observer in any other galaxy, what would you see? Explain using Hubble's law.
  3. A student says "the Earth must be at the centre of the universe because all galaxies recede from us." Critique this argument using the balloon analogy.
Misconceptions, Final Check
Wrong: "Galaxies at the edge of the universe are moving away the fastest because they are being thrown outward from the centre."
Right: There is no centre of expansion. Hubble's law holds from every point in the universe, all observers see all other galaxies receding, with speed proportional to distance. This is a property of uniform expansion, not outward motion from a point.
Wrong: "A galaxy with $z = 2$ is moving at twice the speed of light."
Right: $v = cz$ is only valid for small $z$. For large $z$, the full relativistic cosmological model must be used. Recession velocities exceeding $c$ are possible in general relativity (they reflect the expansion of space itself) but no information travels faster than light.

Three of these statements about Hubble's law are correct. Pick the odd one out.

Analyse7 marks

A graph of recession velocity against distance is plotted for a large sample of galaxies and shows a straight-line trend through the origin with noticeable scatter. Analyse what this graph establishes, including the physical meaning of its gradient and of its scatter. (7 marks)

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Read the shape before reading any number

An analyse answer on a graph starts with the relationship the shape asserts.

The straight line through the origin shows recession velocity is directly proportional to distance, which is Hubble's law, $v = H_0 d$.

Both features earn marks: straight gives proportionality, and through the origin is what makes it direct rather than merely linear.
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Assess6 marks

A student measures a redshift of $z = 0.5$ for a distant galaxy and calculates its recession velocity as $v = cz = 0.5c$, then uses $v = H_0 d$ to find its distance. Assess the validity of this method. (6 marks)

1
Give the method credit where it is sound

An assess answer is a judgement with reasons on both sides. Start with what is right, or the judgement reads as a reflex.

The approach is structurally correct: redshift is the measurable quantity, and converting it to a velocity and then applying Hubble's law is the standard route to a cosmological distance.

Assessing is not fault-finding. Establishing that the strategy is right isolates the problem to one step, which is a sharper criticism.
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Independent practice

Show what you can do without prompts

Complete a shuffled question-bank set, then write full Hubble's law responses in HSC style.

Quick recall, Hubble's law and cosmological redshift
+5 XP

A fresh five-question set drawn from this lesson's bank, feedback shown immediately. +5 XP per correct · +25 XP all correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

Short Answer, 8 marks
+5 XP

ApplyBand 4(3 marks) 1. A galaxy shows an emission line at 680 nm that has rest wavelength 656 nm. (a) Calculate the redshift $z$. (b) Estimate the recession velocity. (c) Estimate the distance using $H_0 = 70$ km/s/Mpc.

1 mark: correct $z$ · 1 mark: correct velocity · 1 mark: correct distance

AnalyseBand 6(5 marks) 2. (a) State Hubble's law and define each term. (b) A galaxy at 200 Mpc is observed. Calculate its recession velocity and redshift. (c) Calculate the scale factor of the universe when the light from a galaxy with $z = 3$ was emitted. (d) Explain why Hubble's law is evidence for an expanding universe rather than galaxies simply moving through a fixed space.

1 mark: state law + define terms · 1 mark: correct $v$ · 1 mark: correct $z$ · 1 mark: correct $a$ · 1 mark: convincing explanation

Show all answers

Multiple choice

MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.

Short Answer, Model Answers

Q1 (3 marks): (a) $z = (680 - 656)/656 = 24/656 = 0.0366$ (1 mark). (b) $v = cz = 3.00\times10^5 \times 0.0366 = 10\,980\ \text{km/s} \approx 1.10\times10^4\ \text{km/s}$ (1 mark). (c) $d = v/H_0 = 10\,980/70 = 157\ \text{Mpc}$ (1 mark).

Q2 (5 marks): (a) Hubble's law: $v = H_0 d$, where $v$ is the recession velocity of a galaxy (km/s), $H_0$ is the Hubble constant ($\approx 70$ km/s/Mpc), and $d$ is the distance of the galaxy (Mpc) (1 mark). (b) $v = 70 \times 200 = 14\,000\ \text{km/s}$; $z = v/c = 14\,000/(3.00\times10^5) = 0.0467$ (1 mark each). (c) $a = 1/(1+z) = 1/(1+3) = 1/4 = 0.25$, the universe was one-quarter its present size (1 mark). (d) Hubble's law shows that recession speed is proportional to distance, this is the hallmark of uniform expansion of space itself, not random motion. An observer anywhere in the universe would observe the same pattern (no special centre), consistent with space stretching uniformly, like dots on an inflating balloon (1 mark).

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Final step

Retrieve, reflect and finish

Check what stuck, revisit your opening redshift prediction and record the method you will reuse.

Check what actually stuck
How did your thinking change?

At the start you were asked to apply Hubble's 1929 law, the linear relationship $v = H_0 d$ that Edwin Hubble derived from 46 galaxy measurements at Mount Wilson Observatory, originally with H₀ ≈ 500 km/s/Mpc (modern value 67.4 km/s/Mpc). The problem used H$\alpha$ shifted from 656.3 nm to 656.5 nm:

  • Did you predict the galaxy is moving away? Correct, the observed wavelength is longer than the rest wavelength, indicating redshift and recession.
  • Did you predict $v \approx cz = (3\times10^8)(0.000305) \approx 9.15\times10^4$ m/s $= 91.5$ km/s? Correct, $z = 0.2/656.3 = 3.05\times10^{-4}$.
  • Did you predict $d = v/H_0 = 91.5/70 \approx 1.31$ Mpc using Hubble's law? Correct, this is roughly the distance to nearby galaxies like M81.

Extend: A galaxy at $z = 0.50$ is observed. Assuming $H_0 = 70$ km/s/Mpc and $c = 3.00\times10^5$ km/s: (a) estimate its recession velocity using $v = cz$, (b) estimate its distance, (c) calculate the scale factor when its light was emitted, and (d) explain why the $v = cz$ formula becomes unreliable at this redshift.

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