Hubble's Law and the Expanding Universe
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.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
A galaxy's hydrogen emission line is observed at 656.5 nm instead of the laboratory value 656.3 nm.
Before reading on, answer:
- Is this galaxy moving toward us or away from us?
- Calculate its recession velocity (use $c = 3.00\times10^8$ m/s).
- 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:
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
- $1 + z = a_{now}/a_{then}$
- 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
Core Content
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.
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.
$v = H_0 d$ — 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)
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:
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 $\approx d/c$.
A distant galaxy shows the H$\alpha$ emission line at 714 nm. The laboratory wavelength is 656 nm.
- Calculate the redshift $z$.
- Estimate the recession velocity.
- Estimate the distance using Hubble's law ($H_0 = 70$ km/s/Mpc).
- Calculate the approximate lookback time.
- Compare this to the age of the universe (~13.8 Gyr).
- Redshift. $$z = \frac{714 - 656}{656} = \frac{58}{656} = \mathbf{0.088}$$
- Recession velocity. $$v = cz = (3.00\times10^5\ \text{km/s})(0.088) = \mathbf{2.64\times10^4\ \text{km/s}}$$
- Distance. $$d = \frac{v}{H_0} = \frac{26400}{70} = \mathbf{377\ \text{Mpc}}$$ $$377\ \text{Mpc} \times 3.26 = \mathbf{1230\ \text{Mly}} \approx 1.23\ \text{Gly}$$
- Lookback time. $$t \approx \frac{d}{c} = \frac{1230}{1000} \approx \mathbf{1.23\ \text{Gyr}}$$
- Comparison. Light left the galaxy when the universe was about $13.8 - 1.2 = 12.6$ Gyr old — about 90% of its current age.
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) $z = (\lambda_{obs}-\lambda_{rest})/\lambda_{rest}$, (2) $v = cz$, (3) $d = v/H_0$, (4) lookback time $\approx d/c$. If $\lambda_{obs} > \lambda_{rest}$ 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.
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:
$$1 + z = \dfrac{a_{now}}{a_{then}} = \dfrac{1}{a_{then}}$$So $a_{then} = 1/(1+z)$. 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.
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.
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.
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 $\lambda_{obs} > \lambda_{rest}$.
Practice using $v = H_0 d$ and the redshift formula
- 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.
- A spectral line at rest wavelength 500 nm is observed at 550 nm. Calculate the redshift, recession velocity, and distance to this galaxy.
- 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).
- Explain why the Hubble time $1/H_0$ overestimates the true age of the universe.
Explain why Hubble's law implies expanding space
- Explain why Hubble's law ($v \propto d$) is evidence that space itself is expanding, rather than galaxies simply flying through a fixed space.
- If you were an observer in any other galaxy, what would you see? Explain using Hubble's law.
- 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.
Three of these statements about Hubble's law are correct. Pick the odd one out.
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.
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
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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).
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.
Five timed questions on Hubble's law and cosmological expansion. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
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