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

Cambridge A‑Level Physics 9702 – Stellar Radii and Cosmic Redshift

1. Introduction

Understanding the sizes of stars and the behaviour of light from distant galaxies are both essential for grasping modern cosmology. This note links the measurement of stellar radii with the observation of redshift, leading to the conclusion that the Universe is expanding.

2. Determining Stellar Radii

The radius \$R\$ of a star can be found using the Stefan‑Boltzmann law combined with the apparent brightness measured from Earth.

\$\$

L = 4\pi R^{2}\sigma T_{\text{eff}}^{4}

\$\$

where:

• \$L\$ – luminosity of the star
• \$\sigma\$ – Stefan‑Boltzmann constant
• \$T_{\text{eff}}\$ – effective surface temperature

Re‑arranging gives:

\$\$

R = \sqrt{\frac{L}{4\pi\sigma T_{\text{eff}}^{4}}}

\$\$

In practice, \$L\$ is obtained from the absolute magnitude \$M\$:

\$\$

M = m - 5\log_{10}\!\left(\frac{d}{10\ \text{pc}}\right)

\$\$

where \$m\$ is the apparent magnitude and \$d\$ the distance in parsecs. Accurate distances are therefore crucial for reliable stellar radii.

3. Redshift – The Key Observation

When light from a distant galaxy is observed, its spectral lines are shifted towards longer wavelengths. The fractional shift is defined as the redshift \$z\$:

\$\$

z = \frac{\lambda{\text{observed}} - \lambda{\text{rest}}}{\lambda_{\text{rest}}}

\$\$

For relatively nearby galaxies (\$z \ll 1\$) the shift can be interpreted as a Doppler effect caused by recession velocity \$v\$:

\$\$

v \approx cz

\$\$

where \$c\$ is the speed of light.

4. Hubble’s Law

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

\$\$

v = H_{0} d

\$\$

Here \$H_{0}\$ is the Hubble constant (approximately \$70\ \text{km s}^{-1}\,\text{Mpc}^{-1}\$). Combining the Doppler approximation with Hubble’s law gives:

\$\$

cz = H{0} d \quad\Longrightarrow\quad z = \frac{H{0}}{c}\,d

\$\$

This equation shows that the farther a galaxy, the larger its redshift, implying a systematic expansion of space itself.

5. Evidence Supporting an Expanding Universe

1. Linear Redshift–Distance Relation: Observations of thousands of galaxies fit the straight line predicted by Hubble’s law.
2. Cosmic Microwave Background: The uniform black‑body radiation is a relic of a hot, dense early Universe that has expanded and cooled.
3. Big Bang Nucleosynthesis: Predicted abundances of light elements match observations, requiring an expanding, cooling Universe.

6. Sample Data Table

7. Connecting Stellar Radii and Redshift

Accurate stellar radii rely on precise distance measurements. For distant galaxies, distances are derived from redshift using Hubble’s law. Thus, the same redshift that signals cosmic expansion also underpins the calculation of stellar sizes beyond our Galaxy.

8. Summary

• Stellar radii are calculated from luminosity, temperature, and distance.
• Redshift \$z\$ quantifies the shift of spectral lines toward longer wavelengths.
• For nearby galaxies, \$z\$ corresponds to a recession velocity via the Doppler approximation.
• Hubble’s law (\$v = H_{0}d\$) demonstrates a proportional increase of velocity with distance, implying that space itself is expanding.
• The expanding‑Universe model provides the framework for converting redshift into distance, which in turn is essential for determining extragalactic stellar radii.