Describe redshift as an increase in the observed wavelength of electromagnetic radiation emitted from receding stars and galaxies

6.2.3 The Universe – Red‑shift

Context within the IGCSE “Space physics” unit

The IGCSE (0625) syllabus covers two main topics in Space physics:

• 6.1 The Earth & the Solar System – rotation, orbit, seasons, Moon phases, planetary order.
• 6.2 The Universe – Red‑shift – the only Universe‑related item, focusing on the expanding Universe.

This note concentrates on the required content for 6.2, but a brief recap of 6.1 is provided so that students can see the whole picture.

Quick Recap – 6.1 The Earth & Solar System

• Earth rotates once every 24 h → day/night cycle.
• Earth orbits the Sun once every 365 days → year; the tilt of its axis (≈ 23.5°) causes the seasons.
• Moon orbits Earth in 27.3 days; phases arise from the changing Sun‑Moon‑Earth geometry.
• Planets orbit the Sun in nearly circular paths; the order from the Sun is Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune.

Learning Objective

Describe red‑shift as an increase in the observed wavelength of electromagnetic radiation emitted by objects that are moving away from the observer, and use this concept to determine recession velocities, distances and the expansion of the Universe.

Key Terminology (syllabus language)

The Doppler Effect for Light

When a light‑emitting object recedes, each successive wave crest is emitted from a position farther away than the previous one. The observer therefore receives a longer wavelength (lower frequency) than was emitted.

Red‑shift definition

\[

z \;=\; \frac{\lambda{\text{obs}}-\lambda{0}}{\lambda_{0}}

\qquad\text{where}

\begin{cases}

\lambda_{0} & = \text{rest (emitted) wavelength}\\[2pt]

\lambda_{\text{obs}} & = \text{wavelength measured by the observer}\\[2pt]

z & = \text{red‑shift (dimensionless)}

\end{cases}

\]

Low‑speed (non‑relativistic) approximation

\[

z \;\approx\; \frac{v}{c}\qquad(v\ll c)

\]

Relativistic red‑shift (required when \(v\gtrsim0.1c\))

\[

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

\qquad\Longleftrightarrow\qquad

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

\]

Measuring Red‑shift – Spectroscopy

1. Record the spectrum of the object with a spectrograph (prism, diffraction grating or CCD‑based instrument).
2. Identify a well‑known spectral line (e.g. H‑α 656.3 nm, Ca K 393.4 nm).
3. Measure its observed wavelength \(\lambda_{\text{obs}}\) on the recorded spectrum.
4. Calculate the red‑shift using \(z = (\lambda{\text{obs}}-\lambda{0})/\lambda_{0}\).
5. Convert \(z\) to a recession velocity with the appropriate (non‑relativistic or relativistic) formula.

Observational Evidence for an Expanding Universe

• Edwin Hubble (1929) measured recession velocities from galaxy spectra and plotted them against distances obtained from Cepheid variables. The points fell on a straight line – the first “Hubble diagram”.
• Modern surveys (e.g. SDSS) contain thousands of galaxies that follow the same linear trend, confirming that space itself is expanding.

Diagram suggestion: a simple plot of velocity (km s⁻¹) on the vertical axis versus distance (Mpc) on the horizontal axis, with a straight line through the origin labelled “Hubble’s law”.

Hubble’s Law – Quantitative Use

Linear form (valid for low‑speed red‑shifts):

\[

v = H_{0}\,d

\]

Combining with the low‑speed red‑shift approximation gives a convenient distance estimate directly from the measured red‑shift:

\[

d \;\approx\; \frac{c}{H_{0}}\,z

\qquad\left(\frac{c}{H_{0}}\approx 4\,300\ \text{Mpc}\right)

\]

Worked Examples

Example 1 – Non‑relativistic case

A spectral line with rest wavelength \(\lambda{0}=500\ \text{nm}\) is observed at \(\lambda{\text{obs}}=510\ \text{nm}\).

1. Red‑shift: \(z = (510-500)/500 = 0.020\).
2. Recession velocity (using \(v \approx zc\)):

   \[

   v = 0.020 \times 3.00\times10^{5}\ \text{km s}^{-1}=6.0\times10^{3}\ \text{km s}^{-1}.

   \]

3. Distance (Hubble’s law):

   \[

   d = \frac{v}{H_{0}} = \frac{6.0\times10^{3}}{70}\ \text{Mpc}\approx 86\ \text{Mpc}.

   \]

Example 2 – Relativistic case (higher red‑shift)

Galaxy with measured \(z = 0.30\).

1. Recession velocity (relativistic formula):

   \[

   v = c\,\frac{(1+0.30)^{2}-1}{(1+0.30)^{2}+1}

   = 3.00\times10^{5}\ \frac{1.69-1}{1.69+1}

   \approx 0.255\,c \approx 7.6\times10^{4}\ \text{km s}^{-1}.

   \]

2. Distance:

   \[

   d = \frac{v}{H_{0}} = \frac{7.6\times10^{4}}{70}\ \text{Mpc}\approx 1.1\times10^{3}\ \text{Mpc}.

   \]

Why Red‑shift Matters

• Provides direct evidence that the Universe is expanding (cosmological red‑shift).
• Allows astronomers to estimate distances to far‑away galaxies using Hubble’s law, which in turn leads to estimates of the age and size of the observable Universe.
• In stellar astronomy, red‑shift (or blueshift) measurements give radial velocities of stars, binary systems and exoplanets.
• Understanding red‑shift underpins technologies that rely on precise knowledge of satellite motion (e.g., GPS timing corrections).

Common Misconceptions

• Red‑shift ≠ “the object looks red”. It is a shift of all wavelengths, not a change in intrinsic colour.
• All red‑shifts are caused by motion. For distant galaxies the dominant cause is the expansion of space (cosmological red‑shift), not ordinary Doppler motion.
• Red‑shift changes brightness. It changes the energy per photon, but the observed flux is primarily governed by distance; the shift itself does not make an object fainter.
• \(z = v/c\) is always accurate. This linear relation fails for \(v\gtrsim0.1c\); the relativistic formula must be used.

Summary Table

Suggested Diagram (to be drawn by the teacher or included in the textbook)

Labelled spectrum showing a known line at its rest wavelength (e.g. H‑α 656.3 nm) and the same line shifted to a longer wavelength for a receding galaxy. An inset can illustrate the “stretching” of successive wave crests as the source moves away.

Questions for Review (AO1 & AO2)

1. Explain why a galaxy moving toward Earth would show a blueshift rather than a red‑shift.
2. A galaxy has a measured red‑shift of \(z = 0.10\).

   • Calculate its recession velocity using the appropriate formula.
   • Estimate its distance using \(H_{0}=70\ \text{km s}^{-1}\text{Mpc}^{-1}\).

3. Discuss how the observation of cosmological red‑shift supports the Big Bang theory.
4. When does the non‑relativistic approximation \(z\approx v/c\) become unreliable? Give a numerical example.
5. Briefly describe one practical application of red‑shift measurements (e.g., determining the motion of satellites for GPS).