Know that redshift in the light from distant galaxies is evidence that the Universe is expanding and supports the Big Bang Theory

6.1 The Earth & the Solar System – Quick Review

Objective

Recall the basic motions of the Earth and the main features of the Solar System that form the foundation for the IGCSE 0625 syllabus (Core).

Key Points

• Earth’s rotation – one complete turn on its axis in 24 h gives a day; the rotation axis is tilted ≈ 23.5° to the orbital plane.
• Earth’s orbit – nearly circular (average radius ≈ 1 AU = 1.496 × 10¹¹ m) with a period of 365.25 days (one year).
• Seasons – result from the axial tilt; different hemispheres receive varying solar angles during the year.
• Moon phases – caused by the changing Sun–Earth–Moon geometry as the Moon orbits Earth (≈ 27.3 days).
• Planetary order (closest to the Sun → farthest): Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune.
• Orbital‑speed formula (circular approximation):

\[

v = \frac{2\pi r}{T}

\]

where r is the orbital radius and T the orbital period.

Worked example – Earth’s orbital speed:

\[

v = \frac{2\pi (1.496\times10^{11}\,\text{m})}{3.156\times10^{7}\,\text{s}}

\approx 2.98\times10^{4}\,\text{m s}^{-1}

\]

6.2 The Universe – Redshift and the Expanding Universe

Objective

Know that red‑shift in the light from distant galaxies is direct evidence that the Universe is expanding and that this observation, together with the CMB and primordial nucleosynthesis, underpins the Big Bang Theory.

Key Concepts

• Light can be shifted toward longer (red) or shorter (blue) wavelengths.
• Red‑shift occurs when the source of light moves away from the observer.
• The dimensionless red‑shift z quantifies the shift.
• For low speeds (< 0.1 c) the Doppler approximation v ≈ cz is adequate; high‑z galaxies require the relativistic Doppler formula.
• Hubble’s Law (v = H₀d) shows recession velocity ∝ distance → space itself is expanding.
• The expanding‑Universe model, together with the CMB and primordial nucleosynthesis, forms the observational basis of the Big Bang Theory.

What is Red‑shift?

When a galaxy recedes, every spectral line it emits is stretched. The red‑shift z is defined as

\[

z = \frac{\Delta\lambda}{\lambda_0}

= \frac{\lambda{\text{obs}} - \lambda0}{\lambda_0}

\]

• λ₀ – laboratory (rest) wavelength of the line.
• λ_obs – wavelength measured in the galaxy’s spectrum.
• z – dimensionless; z > 0 for recession.

Measuring Red‑shift in Practice

1. Obtain a spectrum of the galaxy (e.g., with a diffraction‑grating spectroscope).
2. Identify at least two well‑known lines (e.g., Hα 656.3 nm, Ca K 393.4 nm).
3. Measure the observed wavelength of each line (using a calibrated ruler or software).
4. Calculate z for each line and average the values to reduce random error.

From Red‑shift to Recession Velocity

Non‑relativistic (Doppler) approximation (v ≲ 0.1 c)

\[

v \approx c\,z

\]

Relativistic Doppler formula (required for high‑z galaxies)

\[

1+z = \sqrt{\frac{1+v/c}{1-v/c}}

\;\;\Longrightarrow\;\;

v = c\,\frac{(1+z)^2-1}{(1+z)^2+1}

\]

Both equations use c = 3.00 × 10⁸ m s⁻¹. The relativistic form reduces to v ≈ cz when z ≪ 1.

Hubble’s Law

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

\[

v = H_0 d

\]

• v – recession velocity (m s⁻¹).
• d – distance (usually in megaparsecs; 1 Mpc ≈ 3.09 × 10²² m).
• H₀ – Hubble constant; current best estimate ≈ 70 km s⁻¹ Mpc⁻¹ (± 5 km s⁻¹ Mpc⁻¹).

Observational Evidence for an Expanding Universe

1. Systematic red‑shift – spectra of distant galaxies all show lines shifted to longer wavelengths.
2. Red‑shift–distance relation – plotting v (from red‑shift) against independently measured distances yields a straight line through the origin (Hubble’s diagram).
3. Uniform expansion – the linear relationship implies that space itself, not just the galaxies, is stretching.
4. Cosmological extrapolation – running Hubble’s Law backwards brings all galaxies to a single point ≈ 13.8 Gyr ago, the hot, dense state described by the Big Bang Theory.

Evidence Box 1 – Cosmic Microwave Background (CMB)

The CMB is a uniform 2.73 K black‑body radiation filling space. Discovered by Penzias & Wilson (1965), it is interpreted as relic photons released when the Universe cooled enough for electrons and protons to combine (≈ 380 000 yr after the Big Bang). Its existence is a cornerstone of the Big Bang model.

Evidence Box 2 – Primordial Nucleosynthesis

During the first few minutes of the Universe, nuclear reactions produced light nuclei in the ratios:

These predicted abundances match observations of old, metal‑poor stars and intergalactic gas, providing strong support for a hot early Universe.

Data‑Handling Activity (AO2)

Goal: Use real spectral data to verify Hubble’s Law and practise distance‑indicator concepts.

1. Students receive a table containing:

   • Observed wavelength of the Hα line for several galaxies.
   • Laboratory wavelength (656.3 nm).
   • Independent distance estimates (e.g., from Cepheid‑variable period–luminosity relations or the Tully‑Fisher method). Reminder: Cepheids give distances via the relation \(M = a\log P + b\) where \(P\) is the pulsation period.

2. Calculate z for each galaxy, then the recession velocity using the relativistic formula (or the simple \(v = cz\) if z < 0.1).
3. Plot v (y‑axis) against distance d (x‑axis). Fit a straight line through the origin; the gradient is the experimental Hubble constant.
4. Discuss sources of error (see next section) and compare the experimental H₀ with the accepted value.

Limitations & Sources of Uncertainty

• Peculiar velocities – local gravitational motions (few × 10² km s⁻¹) add scatter, especially for nearby galaxies.
• Instrumental resolution – limited spectral resolution broadens lines, increasing λ_obs uncertainty.
• Calibration errors – inaccurate wavelength calibration leads to systematic shifts.
• Distance‑indicator errors – uncertainties in Cepheid periods, metallicity corrections, or Tully‑Fisher calibrations propagate into the Hubble constant.
• Relativistic effects – using the non‑relativistic formula for high‑z galaxies underestimates v.

Link to the Big Bang Theory (AO1)

The expanding‑Universe model, together with the Cosmic Microwave Background and the primordial nucleosynthesis of light elements, forms the three observational pillars of the Big Bang Theory. By showing that space itself is stretching, that a relic radiation field pervades the cosmos, and that the early Universe produced the observed elemental abundances, these observations provide a coherent, evidence‑based explanation for the origin and evolution of the Universe.

Summary Table

Suggested Diagram

Practical Activity (AO3 – Skills)

Students can perform a simple laboratory experiment using a diffraction grating and a known light source (e.g., a sodium lamp) to record a reference spectrum. By measuring the same lines in a simulated “galaxy” spectrum (produced with a calibrated shift on a computer), they practise:

• Wavelength measurement and error estimation.
• Calculation of red‑shift and recession velocity.
• Construction of a Hubble diagram from the simulated data.

Key Points to Remember

• Red‑shift is a direct observational signature of recession; larger z means a faster‑moving galaxy.
• Hubble’s Law quantifies the expansion and provides a method to estimate cosmological distances.
• Relativistic corrections are essential for high‑z galaxies.
• The expanding‑Universe model, together with the CMB and primordial nucleosynthesis, underpins the Big Bang Theory.
• Understanding uncertainties (peculiar velocities, instrumental errors, distance‑indicator limitations) is crucial for robust data interpretation.