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)

Cambridge IGCSE / A‑Level Physics (9702) – Astrophysics Applications

How this note fits the syllabus

This case‑study links two core syllabus topics to real‑world astrophysics, while explicitly showing connections to the wider Cambridge 9702 programme.

• Topic 7 – Waves and the electromagnetic spectrum: Stefan‑Boltzmann law, black‑body radiation, stellar luminosity.
• Topic 21 – Alternating currents & electromagnetic waves: Doppler shift, red‑shift, recession velocity, Hubble’s law.

All other syllabus units (kinematics, dynamics, work & energy, fields, particles, nuclear physics, etc.) are referenced in the Link‑to‑other‑topics boxes so students can see the wider context.

1. Stellar Radii – Using the Stefan‑Boltzmann Law

1.1 Key definitions (AO1)

1.2 Derivation of the Stefan‑Boltzmann relation (AO1)

For a perfect black‑body the radiant exitance (power per unit area) is

\[

M = \sigma T^{4},

\]

with \$\sigma = 5.670374419\times10^{-8}\ \text{W\,m}^{-2}\text{K}^{-4}\$.

A spherical star of radius \$R\$ has surface area \$A = 4\pi R^{2}\$, therefore

\[

L = A\,M = 4\pi R^{2}\sigma T_{\mathrm{eff}}^{4}.

\]

Re‑arranging for the radius gives the formula used in calculations:

\[

\boxed{R = \sqrt{\dfrac{L}{4\pi\sigma T_{\mathrm{eff}}^{4}}}}.

\]

1.3 Converting observed quantities to \$L\$ and \$T_{\mathrm{eff}}\$ (AO2)

1. Absolute magnitude \$M\$ → luminosity (using the Sun as a reference):

   \[

   L = L{\odot}\,10^{0.4\,(M{\odot}-M)},

   \]

   where \$L{\odot}=3.828\times10^{26}\ \text{W}\$ and \$M{\odot}=4.83\$.

2. Spectral type or colour index → \$T_{\mathrm{eff}}\$. Typical values are listed in the syllabus table for stellar classification (see Link‑to‑Topic 7).

1.4 Uncertainty propagation (AO3)

For \$R = (L/4\pi\sigma T^{4})^{1/2}\$ the fractional uncertainty is

\[

\frac{\delta R}{R}= \frac12\frac{\delta L}{L}+2\frac{\delta T}{T}.

\]

Example: \$\delta L/L = 5\%\$, \$\delta T/T = 2\%\$ → \$\delta R/R = 0.5(0.05)+2(0.02)=0.045\$ (4.5 %).

1.5 Worked example (SI units)

Given: \$M = 0\$, \$T_{\mathrm{eff}} = 10\,000\ \text{K}\$, \$\delta M = 0.1\ \text{mag}\$, \$\delta T = 200\ \text{K}\$.

1. Luminosity:

   \[

   L = 3.828\times10^{26}\ \text{W}\times10^{0.4\,(4.83-0)}

   \approx 1.20\times10^{28}\ \text{W}.

   \]

   Fractional error from magnitude:

   \[

   \frac{\delta L}{L}=0.4\ln10\,\delta M\approx0.921\times0.1=0.092\;(9.2\%).

   \]

2. Radius:

   \[

   R = \sqrt{\frac{1.20\times10^{28}}{4\pi(5.670374419\times10^{-8})(10^{4})^{4}}}

   \approx 1.02\times10^{9}\ \text{m}=1.47\,R_{\odot}.

   \]

3. Uncertainty in \$R\$:

   \[

   \frac{\delta R}{R}= \tfrac12(0.092)+2\left(\frac{200}{10\,000}\right)

   =0.046+0.040=0.086\;(8.6\%),

   \qquad

   \delta R \approx 8.8\times10^{7}\ \text{m}.

   \]

1.6 Link‑to‑other‑topics (quick‑ref)

2. Hubble’s Law – The Expanding Universe

2.1 Doppler (red‑shift) formula (AO1)

For a source moving directly away with speed \$v\ll c\$:

\[

z \equiv \frac{\lambda{\text{obs}}-\lambda{0}}{\lambda_{0}} \approx \frac{v}{c},

\]

where \$z\$ is the (dimensionless) red‑shift and \$c=2.998\times10^{8}\ \text{m s}^{-1}\$.

2.2 Statement of Hubble’s law (vector form, AO1)

\[

\boxed{\mathbf{v}=H_{0}\,\mathbf{d}}

\]

Both \$\mathbf{v}\$ and \$\mathbf{d}\$ point radially away from the observer; \$H_{0}\$ is the Hubble constant (SI unit s⁻¹).

2.3 Converting the conventional value to SI (AO2)

Typical textbook value:

\[

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

\]

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

\[

H_{0}= \frac{70\,000\ \text{m\,s}^{-1}}{3.0857\times10^{22}\ \text{m}}

\approx 2.27\times10^{-18}\ \text{s}^{-1}.

\]

2.4 Step‑by‑step use of Hubble’s law (AO2)

1. Measure the red‑shift \$z\$ from a known spectral line (e.g. Hα at 656.3 nm).
2. Calculate recession speed (valid for \$z\ll1\$):

   \[

   v = cz.

   \]

3. Rearrange Hubble’s law to obtain the distance:

   \[

   d = \frac{v}{H_{0}}.

   \]

4. Insert \$v\$ (m s⁻¹) and \$H_{0}\$ (s⁻¹) → \$d\$ in metres; convert to Mpc if required.

2.5 Worked example (SI units)

Galaxy red‑shift: \$z = 0.015\$.

1. Recession speed:

   \[

   v = cz = (2.998\times10^{8})(0.015)=4.50\times10^{6}\ \text{m s}^{-1}.

   \]

2. Distance:

   \[

   d = \frac{4.50\times10^{6}}{2.27\times10^{-18}}

   =1.98\times10^{24}\ \text{m}\approx 64\ \text{Mpc}.

   \]

3. Uncertainty (assuming \$\delta z = 0.001\$):

   \[

   \frac{\delta v}{v}= \frac{\delta z}{z}= \frac{0.001}{0.015}=0.067,

   \qquad

   \frac{\delta d}{d}= \frac{\delta v}{v}=6.7\%.

   \]

2.6 Expanded practical / experimental component (AO3)

Two complementary activities are suggested so that the case‑study satisfies the required practical skills.

Activity A – Measuring red‑shift with a spectrograph

• Plan: Choose a bright galaxy (e.g. M81), obtain its spectrum with a calibrated diffraction grating spectrograph, record the position of Hα (rest = 656.3 nm).
• Data handling: Convert measured wavelength \$\lambda_{\text{obs}}\$ to \$z\$, then to \$v\$ and \$d\$ using the steps above. Plot \$v\$ versus \$d\$ for at least five galaxies to produce a Hubble diagram.
• Evaluation:

  • Instrumental wavelength calibration (use a neon lamp).
  • Peculiar velocities – estimate their contribution (±300 km s⁻¹) and discuss impact on \$H_{0}\$.
  • Validity of \$v\approx cz\$ – note that for \$z\gtrsim0.1\$ a relativistic formula is required.

Activity B – Laboratory black‑body experiment (link to Topic 7)

• Objective: Verify the Stefan‑Boltzmann law by measuring the power output of a filament lamp at different temperatures.
• Procedure:

  1. Measure voltage \$V\$ and current \$I\$ → electrical power \$P_{\text{elec}} = VI\$.
  2. Use a thermocouple or pyrometer to obtain filament temperature \$T\$.
  3. Place a calibrated radiometer at a known distance \$d\$ to measure radiant flux \$F\$; compute luminosity \$L = 4\pi d^{2}F\$.

• Analysis: Plot \$L\$ against \$T^{4}\$; the gradient should equal \$4\pi\sigma R^{2}\$, allowing an experimental estimate of \$R\$.
• Evaluation:

  • Heat losses by conduction/convection – quantify and discuss.
  • Uncertainty in temperature measurement (thermocouple calibration).
  • Assumption of a perfect black‑body – discuss emissivity corrections.

2.7 Linking Hubble’s law to the Big Bang (AO1 & AO2)

The linear relation \$\mathbf{v}=H_{0}\mathbf{d}\$ implies that every galaxy recedes from every other galaxy. Extrapolating this motion backwards in time reduces all separations to zero at a finite epoch – the Big Bang. Supporting evidence listed in the syllabus includes:

• Cosmic Microwave Background (CMB) – isotropic black‑body radiation (Topic 7).
• Primordial nucleosynthesis – observed abundances of H, He, Li match theoretical predictions (Topic 13).
• Hubble diagram – larger red‑shifts for more distant galaxies confirm an expanding space‑time (Topic 21).

2.8 Link‑to‑other‑topics (quick‑ref)

3. Summary of Key Points for Exam Recall (AO1)

• Stefan‑Boltzmann law: \$L = 4\pi R^{2}\sigma T{\mathrm{eff}}^{4}\$ → \$R = \sqrt{L/(4\pi\sigma T{\mathrm{eff}}^{4})}\$.
• Constants: \$\sigma = 5.670374419\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\$, \$c = 2.998\times10^{8}\ \text{m s}^{-1}\$.
• Red‑shift – velocity relation (for \$z\ll1\$): \$v \approx cz\$.
• Hubble’s law (SI): \$\mathbf{v}=H{0}\mathbf{d}\$ with \$H{0}\approx2.3\times10^{-18}\ \text{s}^{-1}\$.
• Distance from red‑shift: \$d = \dfrac{cz}{H_{0}}\$ (plug in SI values).
• Uncertainty propagation:

  • Radius: \$\displaystyle\frac{\delta R}{R}= \frac12\frac{\delta L}{L}+2\frac{\delta T}{T}\$.
  • Distance: \$\displaystyle\frac{\delta d}{d}= \frac{\delta v}{v}= \frac{\delta z}{z}\$ (if \$H_{0}\$ is taken as exact).

• Big Bang implication – a universal expansion that, when run backwards, leads to a hot, dense singular origin.

4. Useful Physical Constants (SI)

5. Suggested Diagrams for Revision Notes

• Hertzsprung–Russell diagram – shows \$L\propto R^{2}T^{4}\$ and the position of the example star.
• Hubble diagram – recession velocity \$v\$ versus distance \$d\$ with a straight‑line fit defining \$H_{0}\$.
• Red‑shift illustration – laboratory spectrum vs. astronomical spectrum, indicating the wavelength shift.
• Black‑body spectrum – demonstrates the CMB as a perfect black‑body curve.

6. Assessment‑Objective Mapping (AO1–AO3)

7. Quick‑Reference Tables for Other Syllabus Topics Used

7.1 Kinematics & dynamics (Topics 1‑3)

7.2 Work, energy & power (Topic 4)

7.3 Waves & superposition (Topic 5)

7.4 Electricity & DC circuits (Topic 8)

7.5 AC & electromagnetic waves (Topic 21)