Hubble's Law

OCR H556 mapping. This lesson covers the Module 5.5.3 specification points on Hubble's law: $v = H_{0}d$v=H0​d, with the Hubble constant $H_{0} \approx 70$H0​≈70 km s$^{-1}$−1 Mpc$^{-1} \approx 2.3 \times 10^{-18}$−1≈2.3×10−18 s$^{-1}$−1 as the current best estimate; the fact that Hubble's law provides evidence for an expanding universe; and the estimate of the age of the universe as $\approx 1/H_{0} \approx 13.8$≈1/H0​≈13.8 Gyr. It builds directly on the Doppler effect (lesson 7) and sets up the Big Bang theory (lesson 9) and its evidence (lesson 10).

In 1929, the American astronomer Edwin Hubble announced one of the most important discoveries in the history of science. Using the 100-inch telescope at Mount Wilson in California, he had measured the distances to a number of galaxies (using Cepheid variable stars as standard candles) and their radial velocities (using the Doppler effect on their spectral lines). When he plotted velocity against distance, he found a simple linear relationship: more distant galaxies are moving away from us faster.

This deceptively simple observation turned out to mean something extraordinary. The universe is not static. It is expanding. Every point in space is getting farther from every other point, in a uniform and predictable way. Hubble's discovery was the empirical foundation of modern cosmology — and it earned him the lasting honour of having the Hubble Space Telescope named after him.

OCR specification 5.5.3 requires you to know Hubble's law, the Hubble constant, and its connection with the expansion of the universe and the Big Bang. This lesson covers all three.

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Statement of Hubble's law

Hubble's law states that the radial velocity of a galaxy is proportional to its distance from Earth:

$$\boxed{v = H_{0}\,d}$$v=H0​d​

where:

• $v$v is the recession velocity of the galaxy (m s$^{-1}$−1),
• $d$d is the distance from Earth (m),
• $H_{0}$H0​ is the Hubble constant, with units s$^{-1}$−1.

The current best estimate of the Hubble constant is approximately

$$\begin{aligned} H_{0} &\approx 70\,\text{km s}^{-1}\,\text{Mpc}^{-1} \\ &\approx 2.3 \times 10^{-18}\,\text{s}^{-1} \end{aligned}$$H0​​≈70km s−1Mpc−1≈2.3×10−18s−1​

Astronomers usually quote $H_{0}$H0​ in the astronomer-friendly units of km s$^{-1}$−1 per megaparsec — where a megaparsec is $1\,\text{Mpc} \approx 3.09 \times 10^{22}$1Mpc≈3.09×1022 m — but for OCR problems you will often need it in pure SI. Different methods give slightly different values in the range $\approx 67$≈67-$74$74 km s$^{-1}$−1 Mpc$^{-1}$−1 (the Hubble tension, an ongoing puzzle of modern cosmology); $70$70 km s$^{-1}$−1 Mpc$^{-1}$−1 is a reasonable round-figure answer at A-Level, with the SI equivalent $\approx 2.3 \times 10^{-18}$≈2.3×10−18 s$^{-1}$−1. Do not over-precise.

The law holds for galaxies outside our Local Group. Within the Local Group, gravitational interactions dominate over cosmological expansion — Andromeda, for instance, is approaching us despite Hubble's law, because it is gravitationally bound to the Milky Way. Hubble's law applies on the large scale, where random motions and local gravity average out.

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How Hubble discovered it

The chain of reasoning Hubble used was:

1. Measure distances to nearby galaxies using Cepheid variables. A Cepheid is a star whose luminosity varies periodically, with the period tightly correlated with the mean luminosity (the period-luminosity relation discovered by Henrietta Leavitt in 1912). Observing a Cepheid's period gives its absolute brightness; comparing with apparent brightness gives its distance via the inverse-square law (lesson 1).
2. Measure velocities of the same galaxies from their spectral-line shifts, using the Doppler formula $\Delta\lambda/\lambda = v/c$Δλ/λ=v/c (lesson 7).
3. Plot velocity against distance and observe a linear relationship.

Hubble's original data covered galaxies out to about 2 Mpc — not very far by today's standards, but enough to see the trend. Modern measurements, using the Hubble Space Telescope and Type Ia supernovae as standard candles, now extend to distances of several thousand Mpc and confirm the law with high precision.

The constant slope of the velocity-distance plot is, by definition, $H_{0}$H0​. The current best estimate from a range of methods is $H_{0} \approx 70$H0​≈70 km s$^{-1}$−1 Mpc$^{-1}$−1, with the value varying slightly with method — the Hubble tension, an active research question.

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The Hubble constant in SI units

The Hubble constant is quoted astronomically in km s$^{-1}$−1 Mpc$^{-1}$−1 because galactic distances are naturally in megaparsecs and velocities in km s$^{-1}$−1. To convert to pure SI:

$1\,\text{Mpc} = 3.086 \times 10^{22}\,\text{m}$1Mpc=3.086×1022m

So:

$$\begin{aligned} H_{0} &= 70\,\text{km s}^{-1}\,\text{Mpc}^{-1} \\ &= \dfrac{70 \times 10^{3}\,\text{m s}^{-1}}{3.086 \times 10^{22}\,\text{m}} \\ &\approx 2.3 \times 10^{-18}\,\text{s}^{-1} \end{aligned}$$H0​​=70km s−1Mpc−1=3.086×1022m70×103m s−1​≈2.3×10−18s−1​

The units are reciprocal seconds. This is not a coincidence: Hubble's law relates a velocity to a distance, so the constant is a rate — "expansion fractional rate per unit time" — with dimensions of $[\text{time}]^{-1}$[time]−1. The reciprocal,

$$\dfrac{1}{H_{0}} \approx 4.4 \times 10^{17}\,\text{s} \approx 1.4 \times 10^{10}\,\text{yr}$$H0​1​≈4.4×1017s≈1.4×1010yr

is called the Hubble time and is a rough estimate of the age of the universe ($\approx 13.8$≈13.8 Gyr by precise measurements). We return to this below.

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Worked example 1: distance from a red shift

A distant galaxy is observed to have a red shift corresponding to a recession velocity $v = 3.00 \times 10^{6}$v=3.00×106 m s$^{-1}$−1 ($1\%$1% the speed of light). Using $H_{0} = 2.3 \times 10^{-18}$H0​=2.3×10−18 s$^{-1}$−1, find the distance to the galaxy.

$$\begin{aligned} d &= \dfrac{v}{H_{0}} \\ &= \dfrac{3.00 \times 10^{6}}{2.3 \times 10^{-18}} \\ &\approx 1.30 \times 10^{24}\,\text{m} \end{aligned}$$d​=H0​v​=2.3×10−183.00×106​≈1.30×1024m​

Converting to megaparsecs:

$$d \approx \dfrac{1.30 \times 10^{24}}{3.086 \times 10^{22}} \approx 42\,\text{Mpc} \approx 140\,\text{Mly}$$d≈3.086×10221.30×1024​≈42Mpc≈140Mly

This places the galaxy in or beyond the Virgo Cluster — a group of thousands of galaxies $\approx 16$≈16 Mpc away. Sanity check: $v = 3 \times 10^{6}$v=3×106 m s$^{-1}$−1 gives a fractional red shift $z = v/c = 0.01$z=v/c=0.01, so a $1\%$1% wavelength increase. A characteristic H-$\alpha$α line at $656.3$656.3 nm would shift to $\approx 663.0$≈663.0 nm.

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Worked example 2: velocity from a known distance

A galaxy lies at distance $d = 100$d=100 Mpc from Earth. What is its expected recession velocity, assuming Hubble's law holds?

In astronomer units this is trivial:

$$v = H_{0}\,d = 70 \times 100 = 7000\,\text{km s}^{-1}$$v=H0​d=70×100=7000km s−1

In SI:

$$\begin{aligned} d &= 100 \times 3.086 \times 10^{22} = 3.09 \times 10^{24}\,\text{m} \\ v &= H_{0}\,d = (2.3 \times 10^{-18})(3.09 \times 10^{24}) \\ &\approx 7.1 \times 10^{6}\,\text{m s}^{-1} \end{aligned}$$dv​=100×3.086×1022=3.09×1024m=H0​d=(2.3×10−18)(3.09×1024)≈7.1×106m s−1​

$\approx 7000$≈7000 km s$^{-1}$−1, a characteristic velocity for galaxies in the nearby Hubble flow. Note the two forms agree (with rounding) — the astronomer's km s$^{-1}$−1 Mpc$^{-1}$−1 form and the SI form are the same physics expressed in different unit systems.

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Worked example 3: red shift of a distant galaxy

A galaxy lies at distance $d = 500$d=500 Mpc. Find (a) its recession velocity; (b) the observed wavelength of the H-$\alpha$α line (lab wavelength $\lambda = 656.3$λ=656.3 nm) in its spectrum.

(a) Velocity.

$$\begin{aligned} v &= H_{0}\,d = 70 \times 500\,\text{km s}^{-1} \\ &= 3.5 \times 10^{4}\,\text{km s}^{-1} \\ &= 3.5 \times 10^{7}\,\text{m s}^{-1} \end{aligned}$$v​=H0​d=70×500km s−1=3.5×104km s−1=3.5×107m s−1​

This is about $12\%$12% the speed of light — large enough that relativistic corrections start to matter (recall the table in lesson 7), but still in the low-velocity regime to a few per cent.

(b) Observed wavelength. Using the non-relativistic Doppler formula:

$$\begin{aligned} \dfrac{\Delta\lambda}{\lambda} &= \dfrac{v}{c} = \dfrac{3.5 \times 10^{7}}{3 \times 10^{8}} = 0.117 \\ \Delta\lambda &= 0.117 \times 656.3 = 76.5\,\text{nm} \\ \lambda' &= 656.3 + 76.5 = 732.8\,\text{nm} \end{aligned}$$λΔλ​Δλλ′​=cv​=3×1083.5×107​=0.117=0.117×656.3=76.5nm=656.3+76.5=732.8nm​

The H-$\alpha$α line appears at $\approx 733$≈733 nm — well into the deep red of the visible spectrum. A spectrum of this galaxy would show the familiar Balmer lines shifted noticeably towards longer wavelengths. (A proper relativistic calculation would give $\lambda' \approx 738$λ′≈738 nm; the $\approx 0.7\%$≈0.7% correction is at the edge of A-Level visibility but still tolerable.)

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Worked example 4: Hubble time from $H_{0}$H0​ in astronomer units

A textbook quotes $H_{0} = 67.4$H0​=67.4 km s$^{-1}$−1 Mpc$^{-1}$−1 (the Planck 2018 CMB-based value). Calculate the corresponding Hubble time in years.

Step 1. Convert to s$^{-1}$−1:

$$H_{0} = \dfrac{67.4 \times 10^{3}\,\text{m s}^{-1}}{3.086 \times 10^{22}\,\text{m}} = 2.184 \times 10^{-18}\,\text{s}^{-1}$$H0​=3.086×1022m67.4×103m s−1​=2.184×10−18s−1

Step 2. Take the reciprocal:

$$\dfrac{1}{H_{0}} = \dfrac{1}{2.184 \times 10^{-18}} = 4.58 \times 10^{17}\,\text{s}$$H0​1​=2.184×10−181​=4.58×1017s

Step 3. Convert to years (1 yr $= 3.156 \times 10^{7}$=3.156×107 s):

$$\dfrac{1}{H_{0}} \approx \dfrac{4.58 \times 10^{17}}{3.156 \times 10^{7}} \approx 1.45 \times 10^{10}\,\text{yr} \approx 14.5\,\text{Gyr}$$H0​1​≈3.156×1074.58×1017​≈1.45×1010yr≈14.5Gyr

A Hubble time of $\approx 14.5$≈14.5 Gyr — comfortably consistent with the Planck-measured age of the universe of $\approx 13.8$≈13.8 Gyr. The difference comes from the fact that the expansion has not been perfectly constant: deceleration by gravity (for most of cosmic history) and acceleration by dark energy (in the past few Gyr) shift the true age slightly below the Hubble time.

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Hubble's law and the expanding universe

Hubble's observation — that galaxies are receding, and that their speeds are proportional to their distances — points to a strikingly simple conclusion: space itself is expanding.

In an expanding universe, the distance between any two galaxies grows with time. If the expansion is uniform, the rate at which that distance grows is proportional to the distance itself — which is exactly Hubble's law. No galaxy is "the centre" of the expansion; every galaxy sees the same pattern, with nearby galaxies moving slowly and distant ones moving faster.

graph LR
    subgraph "Time t1"
    A1[Galaxy A] -. d .- B1[Galaxy B]
    B1 -. d .- C1[Galaxy C]
    end
    subgraph "Time t2 > t1"
    A2[Galaxy A] -. 2d .- B2[Galaxy B]
    B2 -. 2d .- C2[Galaxy C]
    end

At later time $t_{2}$t2​ every distance has doubled. From A's point of view, B has moved away by $d$d and C by $2d$2d — so C's velocity is twice B's. The same relationship holds from B's or C's point of view. The expansion is isotropic (looks the same in every direction) and homogeneous (looks the same from every point). This is the cosmological principle, which we will meet formally in lesson 10.

A helpful analogy: think of the surface of an inflating balloon. Dots drawn on the balloon are all moving apart, but no dot is "the centre" — the centre of the inflation is not on the surface at all. Every dot sees every other dot moving away, with speed proportional to distance. There is no special dot, and no edge. Our universe behaves similarly (at least to a good approximation on cosmic scales), but in three dimensions rather than two.

graph TD
    BAL["Inflating balloon<br/>(2-D analogue of 3-D space)"]
    BAL --> A["Dots all move apart<br/>as rubber stretches"]
    BAL --> B["Faster apart if<br/>further apart (Hubble's law)"]
    BAL --> C["No 'centre' on the surface<br/>(every dot sees recession)"]
    BAL --> D["Dots do NOT move<br/>across the rubber"]

The balloon analogy clarifies four critical points: distances increase, recession velocity is proportional to distance, no dot is special, and the dots are not moving through the rubber — they are being carried apart by it. The cosmological recession is expansion of space, not motion through space.

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The Hubble time and the age of the universe (revisited)

If the expansion rate has been roughly constant, the time it has taken for any two galaxies to reach their current separation is approximately

$$t = \dfrac{d}{v} = \dfrac{d}{H_{0}\,d} = \dfrac{1}{H_{0}}$$t=vd​=H0​dd​=H0​1​

This ratio $1/H_{0}$1/H0​ is called the Hubble time and gives, as we have already seen, $\approx 14$≈14 Gyr.

This is remarkably close to the precise modern value of $\approx 13.8$≈13.8 Gyr derived from CMB measurements (lesson 10). The Hubble time is an approximation in the sense that it assumes constant expansion. If the expansion has been decelerating (due to gravity, for most of cosmic history), the true age is less than $1/H_{0}$1/H0​. If it has been accelerating (as in the past few Gyr, due to dark energy), the true age is closer to $1/H_{0}$1/H0​. The two corrections nearly cancel, leaving $1/H_{0}$1/H0​ as a few-per-cent-accurate first estimate.

We develop this argument further in lesson 9, where we connect Hubble's law to the idea of a Big Bang.

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