explain why redshift leads to the idea that the Universe is expanding

Cambridge International AS & A Level Physics 9702 – Stellar Radii, Cosmic Red‑shift and the Expanding Universe

Scope & Limitations

This note concentrates on the parts of the Cambridge 9702 syllabus that are directly required for understanding stellar radii, red‑shift and the expanding Universe. The following units are covered (or extended) in the material below:

• Unit 7 – Waves & Light (spectra, Doppler shift)
• Unit 13 – Gravitational Fields (large‑scale cosmology)
• Unit 14 – Temperature & Black‑body Radiation (Stefan‑Boltzmann law)
• Unit 20 – Magnetic Fields (Zeeman effect as a wavelength reference)
• Unit 22 – Quantum Physics (photon energy, E = hc/λ)
• AS & A‑level extensions – Relativity, Cosmology, Hubble’s law, CMB, Big‑Bang nucleosynthesis

The remaining syllabus units (1‑6, 8‑11, 12, 15‑19, 21, 23‑25) are not required for this specific topic but are listed for completeness. A brief “next‑lecture” checklist is provided at the end of the note for teachers who wish to link this material to the broader curriculum.

1. Syllabus Context – Where this material fits

Syllabus Unit	Topic Title	Relevance to this Note
7	Waves – Light and Spectra	Definition of red‑shift, identification of spectral lines, Doppler & relativistic formulas.
13	Gravitational Fields	Cosmological expansion as a large‑scale manifestation of gravity (GR).
14	Temperature – Black‑body Radiation	Stefan‑Boltzmann law for stellar radii; CMB as evidence for expansion.
20	Magnetic Fields – Zeeman Effect	Zeeman splitting provides a laboratory reference for wavelength measurements.
22	Quantum Physics – Photons	Photon energy shift (E = hc/λ) underlies the observed red‑shift.
AS & A‑level extensions	Thermodynamics, Relativity, Cosmology	Relativistic red‑shift, Hubble’s law, observational evidence for the Big Bang.

2. Determining Stellar Radii

2.1. Core equation (Stefan‑Boltzmann law)

The luminosity \(L\) of a star of radius \(R\) and effective temperature \(T_{\rm eff}\) is

\[ L = 4\pi R^{2}\,\sigma T_{\rm eff}^{4}, \qquad \sigma = 5.67\times10^{-8}\ {\rm W\,m^{-2}\,K^{-4}}. \]

Re‑arranged for the radius:

\[ R = \sqrt{\frac{L}{4\pi\sigma T_{\rm eff}^{4}}}. \]

2.2. Obtaining the luminosity

1. Distance‑modulus (units: parsecs) \[ M = m - 5\log_{10}\!\left(\frac{d}{10\ {\rm pc}}\right), \] where \(m\) is the apparent magnitude and \(d\) the distance.
2. From absolute magnitude to luminosity \[ \frac{L}{L_{\odot}} = 10^{0.4\,(M_{\odot}-M)}, \qquad M_{\odot}=4.83. \]

2.3. Uncertainty & systematic effects (AO2 & AO3)

2.4. Links to other syllabus topics

• Unit 14 – Temperature: Direct use of the black‑body law.
• Unit 20 – Magnetic fields: Zeeman splitting confirms the laboratory wavelength \(\lambda_{\rm rest}\) before red‑shift calculation.
• Unit 22 – Quantum physics: Photon energy \(E = hc/\lambda\) changes with red‑shift.

3. Red‑shift – The Key Observation

3.1. Definition

\[ z = \frac{\lambda_{\rm obs}-\lambda_{\rm rest}}{\lambda_{\rm rest}}. \]

3.2. Low‑\(z\) (classical Doppler) approximation

For nearby galaxies (\(z \ll 0.1\)) the shift can be treated as a simple Doppler effect:

\[ v \approx cz,\qquad c = 3.00\times10^{5}\ {\rm km\,s^{-1}}. \]

3.3. Relativistic red‑shift (required for \(z \gtrsim 0.1\))

\[ 1+z = \sqrt{\frac{1+v/c}{\,1-v/c\,}} \;\;\Longrightarrow\;\; v = c\,\frac{(1+z)^{2}-1}{(1+z)^{2}+1}. \]

Use the Doppler form only when \(z<0.05\); otherwise adopt the relativistic expression.

3.4. Practical considerations

• Choose unblended, well‑known laboratory lines (e.g. H α 656.3 nm, Ca II K 393.4 nm).
• Correct for the Earth’s orbital motion (≈ 30 km s⁻¹) and the Sun’s motion relative to the Local Standard of Rest.
• For sources deep in a gravitational potential (e.g. quasars near massive black holes) include the gravitational red‑shift term.

4. Hubble’s Law and the Expanding Universe

4.1. Empirical relationship

Edwin Hubble discovered a linear relation between recession velocity and distance:

\[ v = H_{0}\,d, \qquad H_{0}=70\pm5\ {\rm km\,s^{-1}\,Mpc^{-1}} \;(\text{current best estimate}). \]

4.2. Combining with red‑shift

Insert the appropriate velocity–red‑shift relation (Doppler or relativistic) to obtain the low‑\(z\) approximation:

\[ cz \;(\text{or }v_{\rm rel}) = H_{0}\,d \quad\Longrightarrow\quad z = \frac{H_{0}}{c}\,d. \]

4.3. Physical interpretation – space itself expands

• The linear trend means every galaxy appears to recede from every other galaxy.
• In General Relativity the metric of space stretches; galaxies are carried apart not because they move through space, but because space itself expands.
• Consequences: photon wavelengths are stretched (observed red‑shift) and the cosmic scale factor \(a(t)\) grows with time.

4.4. Suggested diagram

Hubble diagram*: recession velocity (or red‑shift) plotted against distance for thousands of galaxies, showing a straight line through the origin. Outliers (e.g. nearby galaxies with large peculiar motions) are clearly visible.

5. Evidence Supporting an Expanding Universe

1. Linear red‑shift–distance relation – confirmed for > 10 000 galaxies (e.g. SDSS, 2dF surveys).
2. Cosmic Microwave Background (CMB) – near‑perfect black‑body spectrum at 2.73 K; uniformity and anisotropies match predictions of an expanding hot‑big‑bang model.
3. Big‑Bang Nucleosynthesis – predicted primordial abundances of H, He, D, Li agree with spectroscopic measurements.
4. Type Ia Supernova Hubble diagram – distant supernovae appear dimmer than expected in a static universe, indicating accelerated expansion (dark energy).
5. Large‑scale structure & galaxy‑cluster evolution – N‑body simulations require an expanding background to reproduce the observed distribution of galaxies.

6. Sample Data Table (illustrative)

Galaxy / Object	Distance (Mpc)	Observed redshift \(z\)	Recession velocity \(v\) (km s⁻¹)
NGC 224 (Andromeda)	0.78	–0.0010	–300 (approaching)
NGC 7331	14.7	0.0032	960
3C 273 (quasar)	750	0.158	47 400 (relativistic correction ≈ 48 200)
GRB 090423 (high‑z galaxy)	12 500	8.2	2 460 000 (relativistic)

7. Connecting Stellar Radii and Red‑shift

For stars inside the Milky Way, distances are obtained from parallax or main‑sequence fitting. For extragalactic stars (e.g. Cepheid variables in distant galaxies) the distance is inferred from the host galaxy’s red‑shift via Hubble’s law. Consequently:

• Red‑shift provides the distance scale for galaxies.
• Knowing the distance, the absolute magnitude and therefore the luminosity can be calculated.
• Combined with a temperature estimate (from spectral type), the stellar radius follows from the Stefan‑Boltzmann law.

Thus the same observational phenomenon that underpins the expanding‑Universe model is essential for measuring stellar properties beyond the Milky Way.

8. Summary – Key points for exam (AO1, AO2, AO3)

• Stellar radius: \(R = \sqrt{L/(4\pi\sigma T_{\rm eff}^{4})}\); \(L\) derived from the distance‑modulus and bolometric correction.
• Red‑shift: \(z = (\lambda_{\rm obs}-\lambda_{\rm rest})/\lambda_{\rm rest}\). Use \(v\approx cz\) for \(z<0.05\); otherwise apply the relativistic formula.
• Hubble’s law: \(v = H_{0}d\) → linear red‑shift–distance relation, the observational basis for the statement “space itself expands”.
• Uncertainty analysis: propagate errors in \(m\), \(d\), \(T_{\rm eff}\) to obtain \(\delta R\); recognise systematic effects (extinction, bolometric correction, Zeeman reference).
• Evidence for expansion: Hubble diagram, CMB, nucleosynthesis, Type Ia supernovae, large‑scale structure.
• Cross‑topic links: temperature (black‑body), quantum (photon energy), magnetic fields (Zeeman), gravity (cosmology) – all appear in the syllabus and reinforce the concepts.

9. Checklist of Uncovered Syllabus Units (for future lessons)

The following units are not required for the present topic but form part of the full 9702 syllabus. Teachers may wish to schedule separate sessions to address them.

• Units 1‑6 – Physical quantities, Kinematics, Dynamics, Forces, Work & Energy, Deformation.
• Unit 8 – Superposition and interference (wave phenomena).
• Units 9‑11 – Oscillations, Mechanical waves, Sound.
• Unit 12 – Nuclear physics (radioactivity, decay).
• Units 15‑19 – Thermal physics, Ideal gases, Kinetic theory, Thermodynamics, Fluid dynamics.
• Unit 21 – Atomic structure (energy levels, spectra beyond hydrogen).
• Units 23‑25 – Advanced topics: particle physics, nuclear reactions, modern applications.