Know that the speed v at which a galaxy is moving away from the Earth can be found from the change in wavelength of the galaxy's starlight due to redshift

6.1 The Earth and the Solar System

6.1.1 Earth’s rotation and day‑night cycle

• Earth rotates once about its axis every 24 h (≈ 86 400 s).
• Rotation causes the apparent daily motion of the Sun and stars – the basis of the day‑night cycle.
• Angular speed: \(\displaystyle \omega = \frac{2\pi}{T}= \frac{2\pi}{24\ \text{h}} \approx 7.3\times10^{-5}\ \text{rad s}^{-1}\).

6.1.2 Earth’s revolution and the seasons

• Earth orbits the Sun once every 365 days (≈ 3.16 × 10⁷ s).
• The orbital plane is inclined by 23.5° to the Sun‑Earth line; this tilt produces the seasons.
• Mean orbital speed (circular approximation):

  \[

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

  \approx \frac{2\pi(1.5\times10^{11}\ \text{m})}{3.16\times10^{7}\ \text{s}}

  \approx 30\ \text{km s}^{-1}.

  \]

6.1.3 The Moon’s orbit and phases

• Moon revolves around Earth in about 27.3 days (sidereal month).
• Because the Moon reflects sunlight, its visible shape changes – the lunar phases.
• Average orbital speed: \(\displaystyle v \approx 1.0\ \text{km s}^{-1}\).

6.1.4 Basic layout of the Solar System

AU = astronomical unit ≈ 1.5 × 10⁸ km (average Earth‑Sun distance).

6.1.5 Quick quantitative example

Calculate the linear speed of Earth in its orbit.

\[

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

= \frac{2\pi(1.5\times10^{11}\ \text{m})}{3.16\times10^{7}\ \text{s}}

\approx 30\ \text{km s}^{-1}.

\]

6.2 Supplementary/Extension – Expanding Universe, Red‑shift and Galaxy Velocities

6.2.1 The expanding Universe (conceptual)

• Observations show that, on large scales, galaxies are moving away from each other – the Universe is expanding.
• The expansion is described by a single number, the Hubble constant \(H_{0}\). For IGCSE work we use

  \[

  H_{0}\approx 70\ \text{km s}^{-1}\,\text{Mpc}^{-1}.

  \]

6.2.2 Hubble’s Law

The linear relationship between a galaxy’s recession speed \(v\) and its distance \(d\) is

\[

v = H_{0}\,d

\]

• \(v\) – recession velocity (km s\(^{-1}\)).
• \(d\) – distance from Earth (megaparsecs, Mpc; 1 Mpc ≈ 3.09 × 10¹⁹ km).
• Works well for relatively nearby galaxies (typically \(d \lesssim 200\) Mpc).

6.2.3 Red‑shift – definition and linear Doppler approximation

When a light source recedes, its wavelength is stretched. The red‑shift \(z\) is defined as

\[

z = \frac{\Delta\lambda}{\lambda_{0}}

= \frac{\lambda{\text{obs}}-\lambda{0}}{\lambda_{0}},

\]

where

• \(\lambda_{0}\) – rest (laboratory) wavelength of a known spectral line.
• \(\lambda_{\text{obs}}\) – wavelength measured in the galaxy’s spectrum.
• \(z>0\) indicates recession (red‑shift); \(z<0\) would indicate approach (blue‑shift).

For the speeds encountered in IGCSE examinations (\(v \ll c\)), the Doppler shift reduces to the simple linear form

\[

\boxed{v \approx c\,z = c\,\frac{\Delta\lambda}{\lambda_{0}}}

\]

with \(c = 3.00\times10^{8}\ \text{m s}^{-1}\).

Advanced note (optional): For very distant galaxies (\(v\) approaching a significant fraction of \(c\)) the relativistic formula

\(1+z = \sqrt{\frac{1+v/c}{1-v/c}}\) is required, but this is beyond the IGCSE syllabus.

6.2.4 Determining a galaxy’s recession velocity

1. Identify a clear spectral line with a known rest wavelength \(\lambda_{0}\) (e.g., Hα = 656.3 nm).
2. Measure the observed wavelength \(\lambda_{\text{obs}}\) from the galaxy’s spectrum.
3. Calculate the shift: \(\Delta\lambda = \lambda{\text{obs}}-\lambda{0}\).
4. Find the red‑shift: \(z = \Delta\lambda/\lambda_{0}\).
5. Convert to velocity using the linear relation \(v = c\,z\). (Only this step is required for IGCSE.)

6.2.5 Estimating the galaxy’s distance

Combine the measured velocity with Hubble’s law:

\[

d = \frac{v}{H{0}} = \frac{c\,z}{H{0}}.

\]

Insert \(c = 3.00\times10^{5}\ \text{km s}^{-1}\) and \(H_{0}=70\ \text{km s}^{-1}\,\text{Mpc}^{-1}\) for a quick distance estimate.

6.2.6 Typical spectral lines used in red‑shift work

6.2.7 Worked example – from red‑shift to distance

In a galaxy the Hα line is observed at \(\lambda_{\text{obs}} = 720.0\ \text{nm}\).

1. Rest wavelength: \(\lambda_{0}=656.3\ \text{nm}\).
2. Shift: \(\Delta\lambda = 720.0 - 656.3 = 63.7\ \text{nm}\).
3. Red‑shift: \(z = \dfrac{63.7}{656.3} = 0.097\).
4. Recession velocity (linear formula):

   \[

   v = c\,z = (3.00\times10^{5}\ \text{km s}^{-1})(0.097) \approx 2.9\times10^{4}\ \text{km s}^{-1}.

   \]

5. Distance using Hubble’s law:

   \[

   d = \frac{v}{H_{0}} = \frac{2.9\times10^{4}\ \text{km s}^{-1}}{70\ \text{km s}^{-1}\,\text{Mpc}^{-1}} \approx 414\ \text{Mpc}.

   \]

6. In light‑years: \(1\ \text{Mpc} \approx 3.26\times10^{6}\) ly, so the galaxy is about \(1.35\times10^{9}\) ly away.

6.2.8 Quick revision checklist

• Red‑shift definition: \(z = \Delta\lambda/\lambda_{0}\).
• Linear Doppler relation for IGCSE: \(v = c\,z\) (valid when \(v \ll c\)).
• Hubble’s law: \(v = H{0}d\) with \(H{0}\approx70\ \text{km s}^{-1}\,\text{Mpc}^{-1}\).
• Procedure to go from an observed spectral line to a galaxy’s distance.
• Remember the limits: use the linear formula only for modest red‑shifts (typically \(z \lesssim 0.1\)).