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

6.2.3 The Universe – Redshift and Galaxy \cdot elocities

Key Concept

The speed \(v\) at which a galaxy is moving away from the Earth can be determined from the change in wavelength of the galaxy’s starlight due to redshift. For velocities that are small compared with the speed of light, the relationship is linear:

\$\$

v = c \,\frac{\Delta \lambda}{\lambda_0}

\$\$

where

• \(c\) is the speed of light (\(3.00\times10^8\ \text{m s}^{-1}\)),
• \(\lambda_0\) is the rest wavelength of the spectral line (the wavelength emitted by the source when it is at rest relative to the observer),
• \(\Delta \lambda = \lambda{\text{obs}} - \lambda0\) is the observed shift in wavelength.

Steps to Calculate Galaxy \cdot elocity

1. Identify a spectral line in the galaxy’s spectrum that has a known rest wavelength \(\lambda_0\).
2. Measure the observed wavelength \(\lambda_{\text{obs}}\) of that line in the galaxy’s spectrum.
3. Calculate the wavelength shift: \(\Delta \lambda = \lambda{\text{obs}} - \lambda0\).
4. Insert the values into the formula \(v = c \,\frac{\Delta \lambda}{\lambda_0}\) to obtain the recession velocity \(v\).
5. Check the result: if \(v\) is much less than \(c\) (typically \(v < 0.1c\)), the linear approximation is valid. For larger velocities, use the relativistic Doppler formula.

Example Calculation

Suppose the hydrogen Balmer line Hα has a rest wavelength \(\lambda0 = 656.3\ \text{nm}\). In the spectrum of a distant galaxy, this line is observed at \(\lambda{\text{obs}} = 700.0\ \text{nm}\).

• Wavelength shift: \(\Delta \lambda = 700.0\ \text{nm} - 656.3\ \text{nm} = 43.7\ \text{nm}\).
• Convert to metres: \(\Delta \lambda = 43.7 \times 10^{-9}\ \text{m}\), \(\lambda_0 = 656.3 \times 10^{-9}\ \text{m}\).
• Velocity:

  \$\$

  v = (3.00\times10^8\ \text{m s}^{-1}) \times \frac{43.7\times10^{-9}\ \text{m}}{656.3\times10^{-9}\ \text{m}}

  = 2.00\times10^7\ \text{m s}^{-1}

  \$\$

• Convert to km s⁻¹: \(v \approx 20\,000\ \text{km s}^{-1}\).

Typical Spectral Lines Used in Redshift Studies

Notes for the Exam

• Remember that redshift is always positive for galaxies moving away from us (wavelengths are stretched).
• Use the linear formula only when \(v \ll c\). For velocities approaching a significant fraction of the speed of light, the relativistic Doppler shift must be applied:

\$\$

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

\$\$

where \(z = \Delta \lambda / \lambda_0\).

• When calculating, keep units consistent (metres for wavelengths, metres per second for velocity).
• Check that the result is reasonable: typical recession velocities for nearby galaxies are a few thousand km s⁻¹, while very distant galaxies can have velocities of tens of thousands of km s⁻¹.