recall and use Hubble’s law v . H0d and explain how this leads to the Big Bang theory (candidates will only be required to use SI units)

Stellar Radii

In A‑Level physics the radius of a star, \$R\$, is a fundamental parameter that can be related to observable quantities such as luminosity \$L\$ and effective temperature \$T_{\mathrm{eff}}\$ via the Stefan‑Boltzmann law:

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

where \$\sigma = 5.670374419\times10^{-8}\ \text{W\,m}^{-2}\text{K}^{-4}\$ is the Stefan‑Boltzmann constant. Rearranging gives an expression for the radius:

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

Steps to calculate a stellar radius

1. Obtain the star’s luminosity \$L\$ in watts (SI unit). For distant stars this is often derived from the absolute magnitude \$M\$ using \$L = L{\odot}\,10^{0.4\,(M{\odot}-M)}\$ where \$L{\odot}=3.828\times10^{26}\ \text{W}\$ and \$M{\odot}=4.83\$.
2. Determine the effective temperature \$T_{\mathrm{eff}}\$ in kelvin (K) from spectral classification or colour indices.
3. Insert \$L\$ and \$T_{\mathrm{eff}}\$ into the radius formula above.
4. Express the result in metres (m) or, more conveniently, in solar radii \$R_{\odot}=6.96\times10^{8}\ \text{m}\$.

Example calculation

Consider a star with \$M = 0\$ and \$T_{\mathrm{eff}} = 10\,000\ \text{K}\$.

• Luminosity: \$L = 3.828\times10^{26}\ \text{W}\times10^{0.4\,(4.83-0)} \approx 1.2\times10^{28}\ \text{W}\$
• Radius: \$R = \sqrt{\frac{1.2\times10^{28}}{4\pi(5.670374419\times10^{-8})(10^{4})^{4}}}\approx 1.0\times10^{9}\ \text{m}\approx1.44\,R_{\odot}\$

Hubble’s Law and the Expanding Universe

Hubble’s law relates the recession velocity \$v\$ of a distant galaxy to its distance \$d\$ from us:

\$v = H_{0}\,d\$

where \$H_{0}\$ is the Hubble constant, typically expressed in units of \$\text{km\,s}^{-1}\text{Mpc}^{-1}\$. For A‑Level work we convert to SI units (metres per second per metre):

\$1\ \text{Mpc}=3.0857\times10^{22}\ \text{m}\$

\$H_{0}=70\ \text{km\,s}^{-1}\text{Mpc}^{-1}=70\,000\ \text{m\,s}^{-1}\big/3.0857\times10^{22}\ \text{m}\approx2.27\times10^{-18}\ \text{s}^{-1}\$

Using Hubble’s law

1. Measure the redshift \$z\$ of a galaxy’s spectral lines.
2. For \$z\ll1\$, approximate the recession velocity as \$v \approx cz\$, where \$c=2.998\times10^{8}\ \text{m\,s}^{-1}\$.
3. Rearrange Hubble’s law to find the distance: \$d = \frac{v}{H_{0}}\$
4. Insert \$v\$ and \$H_{0}\$ in SI units to obtain \$d\$ in metres.

Example

A galaxy shows a redshift \$z = 0.01\$.

• Recession velocity: \$v \approx cz = (2.998\times10^{8}\ \text{m\,s}^{-1})(0.01)=2.998\times10^{6}\ \text{m\,s}^{-1}\$.
• Distance: \$d = \frac{2.998\times10^{6}}{2.27\times10^{-18}} \approx 1.32\times10^{24}\ \text{m} \approx 42.8\ \text{Mpc}\$

Link to the Big Bang Theory

The linear relationship \$v = H_{0}d\$ implies that every galaxy is moving away from every other galaxy, suggesting that the Universe was once much denser and hotter. Extrapolating the expansion backwards in time leads to a singular origin – the Big Bang.

• Uniform expansion → isotropic cosmic microwave background (CMB).
• Predicted primordial nucleosynthesis matches observed light‑element abundances.
• Redshift observations confirm that distant galaxies are receding faster, consistent with an expanding space‑time.

Key points for exam recall

• Hubble’s law: \$v = H{0}d\$ (SI units: \$\text{m\,s}^{-1}\$ for \$v\$, \$\text{m}\$ for \$d\$, \$\text{s}^{-1}\$ for \$H{0}\$).
• Typical value of \$H_{0}\$: \$70\ \text{km\,s}^{-1}\text{Mpc}^{-1}\approx2.3\times10^{-18}\ \text{s}^{-1}\$.
• Stellar radius from luminosity and temperature: \$R = \sqrt{L/(4\pi\sigma T_{\mathrm{eff}}^{4})}\$.
• Big Bang inference: a universal expansion implies a finite age and an initial hot, dense state.

Useful constants (SI)