Hubble's Law

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 further 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:

v = H_0 × d

where:

• `v) is the recession velocity of the galaxy (in m s⁻¹),
• `d) is the distance from Earth (in m),
• `H_0) is the Hubble constant, with units of s⁻¹.

The modern value of the Hubble constant is approximately

H_0 ≈ 70 km s⁻¹ Mpc⁻¹
   ≈ 2.27 × 10⁻¹⁸ s⁻¹

(Astronomers commonly quote H_0) in the astronomer-friendly units of km/s per megaparsec — where a megaparsec is approx 3.09 × 10²²) m — but for OCR problems you will often need it in pure SI: `H_0 approx 2 × 10⁻¹⁸) s⁻¹.)

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.
2. Measure velocities of the same galaxies from their spectral line shifts, using the Doppler formula `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 supernovae Ia as standard candles, now extend to distances of several thousand Mpc and confirm the law with high precision.

graph LR
    A[Galaxy 1<br/>d small<br/>v small] --> L[Velocity-Distance<br/>plot]
    B[Galaxy 2<br/>d medium<br/>v medium] --> L
    C[Galaxy 3<br/>d large<br/>v large] --> L
    L --> S[Straight line<br/>v = H₀ d]

The constant slope of the velocity–distance plot is, by definition, H_0). Different teams, using different standard candles, have arrived at values in the range 67 ext{–}74) km s⁻¹ Mpc⁻¹. The fact that this number varies slightly depending on the method — the famous Hubble tension — is one of the hottest questions in modern cosmology, but is beyond the scope of A-Level.

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The Hubble Constant In SI Units

The Hubble constant is quoted astronomically in km s⁻¹ Mpc⁻¹ because galactic distances are naturally in megaparsecs and velocities in km/s. To convert to pure SI units:

1 Mpc = 3.086 × 10²² m

So:

H_0 = 70 km s⁻¹ Mpc⁻¹
    = 70 × 10³ m s⁻¹ / (3.086 × 10²² m)
    = 2.27 × 10⁻¹⁸ s⁻¹

The units are reciprocal seconds. This is not a coincidence: Hubble's law is a relationship between a velocity and a distance, so the constant is a rate — "expansion rate per unit length per unit time" — with dimensions of `[1/T]). The reciprocal,

1/H_0 ≈ 4.4 × 10¹⁷ s ≈ 1.4 × 10¹⁰ yr

is called the Hubble time and is a rough estimate of the age of the universe. We will return to this in Lesson 9.

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Worked Example 1: Distance From A Redshift

A distant galaxy is observed to have a red shift corresponding to a recession velocity of v = 3.00 × 10⁶) m s⁻¹ (1% the speed of light). Using H_0 = 2.27 × 10⁻¹⁸) s⁻¹, find the distance to the galaxy.

d = v / H_0
  = (3.00 × 10⁶) / (2.27 × 10⁻¹⁸)
  = 1.32 × 10²⁴ m

Converting to megaparsecs:

d = (1.32 × 10²⁴) / (3.086 × 10²²)
  ≈ 43 Mpc ≈ 140 million light years

This would place the galaxy in or beyond the Virgo Cluster — a group of thousands of galaxies about 16 Mpc away. A quick sanity check: v = 3 × 10⁶) m/s means a fractional red shift z = v/c = 0.01), i.e. a 1% wavelength increase. A characteristic H-α line at 656.3) nm would shift to about 663.0) nm.

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Worked Example 2: Velocity From A Known Distance

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

d = 100 × 3.086 × 10²² = 3.086 × 10²⁴ m
v = H_0 × d
  = (2.27 × 10⁻¹⁸) × (3.086 × 10²⁴)
  = 7.00 × 10⁶ m s⁻¹

In km/s, that is 7000 km s⁻¹, a characteristic velocity for galaxies in the "nearby" Hubble flow. In more familiar astronomer units, this is the same as using H_0 = 70) km/s/Mpc directly: v = 70 × 100 = 7000) km/s.

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Worked Example 3: Redshift Of A Distant Galaxy