Hubble's law and the Big Bang theory
Section: Astronomy and Cosmology | Syllabus: Cambridge A Level Physics (9702)
Hubble's Law and the Big Bang Theory
Redshift of Spectral Lines
When we observe the spectra of distant galaxies, we find that the emission and absorption lines are shifted from their known laboratory values.
ObservationThe spectral lines from almost all distant galaxies show an increase in wavelength compared to their known values. This shift towards longer wavelengths is called redshift.
Figure: Redshifted SpectrumComparison of laboratory hydrogen spectrum with the observed spectrum from a distant galaxy. All lines are shifted to longer wavelengths (towards the red end).
The Doppler Effect for Light
The redshift is caused by the Doppler effect: when a source moves away from an observer, the observed wavelength increases (and frequency decreases).
(Δ λ/λ) ≈ (Δ f/f) ≈ (v/c)where Δ λ = change in wavelength, λ = original wavelength, Δ f = change in frequency, f = original frequency, v = recession velocity, c = speed of light Sign Convention
- Redshift (Δλ > 0): source moving away from observer
- Blueshift (Δλ < 0): source moving towards observer
The redshift parameter z = (Δλ/λ) is positive for recession.
Evidence for an Expanding Universe
Key Observations- Almost all galaxies show redshift (moving away from us)
- More distant galaxies have greater redshifts (moving away faster)
- This pattern is observed in all directions
Conclusion: The redshift of distant galaxies provides evidence that the Universe is expanding. All galaxies are moving away from each other - not because they are moving through space, but because space itself is expanding.
Important ClarificationThe expansion does not mean Earth is at the centre of the Universe. An observer in any galaxy would see the same pattern - all other galaxies moving away, with more distant ones receding faster.
Hubble's Law
In 1929, Edwin Hubble discovered a relationship between the recession velocity of galaxies and their distance.
Hubble's Law The recession velocity of a galaxy is directly proportional to its distance from us. v ≈ H_0 dwhere v = recession velocity (m s^-1), H_0 = Hubble constant, d = distance (m)
H_0 ≈ 2.2 × 10^-18 s^-1 (in SI units) Figure: Hubble Diagram
A graph of recession velocity vs distance for galaxies shows a straight line through the origin. The gradient equals the Hubble constant H_0.
The Big Bang Theory
Hubble's law leads directly to the Big Bang theory of the origin of the Universe.
ReasoningIf all galaxies are moving apart now, then in the past they must have been closer together. Running time backwards:
- All matter was once concentrated in a single point
- The Universe began with an explosive expansion - the Big Bang
- Space, time, matter and energy all originated from this event
Estimating the Age of the Universe
The Hubble constant can be used to estimate how long ago the Big Bang occurred.
Age of Universe ≈ (1/H_0)(1/H_0) = 12.2 × 10^-18 ≈ 4.5 × 10^17 s ≈ 14 billion years Evidence Supporting the Big Bang
- Redshift of galaxies: Shows the Universe is expanding
- Cosmic Microwave Background (CMB): Remnant radiation from the early Universe
- Abundance of light elements: Proportions of hydrogen and helium match predictions
Worked Examples
Worked Example: Calculating Recession VelocityQuestion: A spectral line normally at 600.00 nm is observed at 600.80 nm in a star's spectrum. Calculate the star's velocity relative to Earth.
SolutionΔλ = 600.80 - 600.00 = 0.80 nm
v = (Δλ/λ) × c = (0.80/600.00) × 3.00 × 10^8
v = 1.33 × 10^-3 × 3.00 × 10^8 = 4.0 × 10^5 m s^-1
Since wavelength increased (redshift), the star is moving away from Earth.
Worked Example: Using Hubble's LawQuestion: A galaxy has a recession velocity of 1.5 × 10^7 m s^-1. Calculate its distance from Earth. (H_0 = 2.2 × 10^-18 s^-1)
Solutiond = (v/H_0) = 1.5 × 10^72.2 × 10^-18
d = 6.8 × 10^24 m (about 720 million light years)
Summary of Key Equations
| Quantity | Equation | Use |
|---|---|---|
| Radiant flux intensity | F = (L/4π d^2) | Finding distance using standard candles |
| Wien's displacement law | _max ∝ (1/T) | Finding surface temperature of stars |
| Stefan-Boltzmann law | L = 4πσ r^2 T^4 | Finding stellar radius |
| Doppler redshift | (Δλ/λ) ≈ (v/c) | Finding recession velocity |
| Hubble's law | v ≈ H_0 d | Relating velocity to distance |
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