use the Stefan–Boltzmann law L = 4πσr 2 T

Stellar Radii – Using the Stefan–Boltzmann Law

Learning Objectives (AO1‑AO3)

• AO1: State the Stefan–Boltzmann law, define each symbol and explain the physical meaning of the terms.
• AO2: Manipulate the law to calculate a star’s radius when its luminosity and effective temperature are given, including unit conversion and error propagation.
• AO3: Analyse how luminosity and temperature influence radius, evaluate the limits of the black‑body model (emissivity, spectral deviations) and discuss uncertainties.

Syllabus Mapping (Cambridge International AS & A Level Physics 9702)

Physics Foundations – Quantities & Units

The Stefan–Boltzmann Law (Ideal Black‑Body)

The total radiant power emitted by a spherical black‑body of radius r and temperature T is

\$ L = 4\pi \sigma r^{2} T^{4} \$

• L – luminosity (W)
• r – radius of the radiating surface (m)
• σ – Stefan–Boltzmann constant = 5.670 × 10⁻⁸ W m⁻² K⁻⁴
• T – effective surface temperature (K)

The factor 4πr² is the surface area of a sphere, reflecting isotropic emission.

Brief Derivation (link to Topic 22)

Integrating Planck’s spectral radiance B(ν,T) over all frequencies (0 → ∞) and over the full solid angle (4π sr) yields

\$ \int_{0}^{\infty}\!\!B(\nu,T)\,d\nu = \sigma T^{4} \$

where

\$ \sigma = \frac{2\pi^{5}k^{4}}{15c^{2}h^{3}} \$

showing that σ is built from the Boltzmann constant k, Planck’s constant h and the speed of light c.

Real‑Star Corrections (Emissivity)

Real stars are not perfect black‑bodies; their surface emissivity ε is slightly less than 1. The more general form is

\$ L = 4\pi \sigma \, \varepsilon \, r^{2} T^{4} \$

• For most main‑sequence stars, ε ≈ 0.9 – 1, so the black‑body approximation is acceptable for exam‑level work.
• If a problem supplies ε, simply replace σ with εσ in all calculations.

Rearranging for Stellar Radius

Solving for r (including emissivity) gives

\$ r = \sqrt{\frac{L}{4\pi \sigma \varepsilon T^{4}}} \$

Key proportionalities (ε ≈ 1):

• r ∝ √L (a ten‑fold increase in luminosity → radius ≈ 3.2 times larger, if T constant)
• r ∝ 1/T² (doubling T → radius reduces by a factor of 4, if L constant)

Error Propagation (AO3)

When L and T carry uncertainties (ΔL, ΔT), the fractional uncertainty in r is

\$ \frac{\Delta r}{r} = \frac{1}{2}\frac{\Delta L}{L} + 2\frac{\Delta T}{T} \$

(ε is usually taken as exact for exam purposes.) This relationship highlights why temperature measurements dominate the error budget.

Step‑by‑Step Procedure (AO2)

1. Record the given data. Convert any non‑SI values to SI:

   • 1 L⊙ = 3.828 × 10²⁶ W
   • 1 R⊙ = 6.96 × 10⁸ m
   • If emissivity is supplied, note ε; otherwise set ε = 1.

2. Calculate T⁴. Use scientific‑notation mode on the calculator.
3. Form the denominator

   \$ D = 4\pi \sigma \varepsilon T^{4} \$

4. Form the ratio

   \$ Q = \frac{L}{D} \$

5. Take the square root**

   \$ r = \sqrt{Q} \$

6. Express the result** in metres, then optionally in solar radii:

   \$ r{(R\odot)} = \frac{r}{6.96\times10^{8}\ \text{m}} \$

7. Propagate uncertainties** using the formula above, if ΔL and ΔT are given.

Worked Example – The Sun (AO1‑AO3)

Given:

• L⊙ = 3.828 × 10²⁶ W
• T⊙ = 5 778 K
• ε = 1 (perfect black‑body approximation)

Solution:

1. T⁴ = (5.778 × 10³ K)⁴ = 1.11 × 10¹⁵ K⁴
2. Denominator

   \$ D = 4\pi\sigma T^{4}=4\pi(5.670\times10^{-8})(1.11\times10^{15})=7.93\times10^{8}\ \text{W m}^{-2} \$

3. Ratio

   \$ Q = \frac{L}{D}= \frac{3.828\times10^{26}}{7.93\times10^{8}}=4.83\times10^{17}\ \text{m}^{2} \$

4. Radius

   \$ r = \sqrt{4.83\times10^{17}} = 6.96\times10^{8}\ \text{m}=1.00\ R_{\odot} \$

5. Uncertainty (illustrative) – if ΔL/L = 2 % and ΔT/T = 1 %:

   \$ \frac{\Delta r}{r}= \frac{1}{2}(0.02)+2(0.01)=0.02\;(2\%) \$

Worked Example – Sirius A (Main‑Sequence Star)

Data:

• L = 25.4 L⊙
• T = 9 940 K
• ε ≈ 0.98 (typical for an A‑type star; if not given, use ε = 1)

Solution:

1. L = 25.4 × 3.828 × 10²⁶ W = 9.72 × 10²⁷ W
2. T⁴ = (9.94 × 10³ K)⁴ = 9.77 × 10¹⁵ K⁴
3. D = 4πσ ε T⁴ = 4π(5.670 × 10⁻⁸)(0.98)(9.77 × 10¹⁵) = 6.83 × 10⁹ W m⁻²
4. Q = L/D = 9.72 × 10²⁷ / 6.83 × 10⁹ = 1.42 × 10¹⁸ m²
5. r = √Q = 1.19 × 10⁹ m = 1.71 R⊙

Typical Stellar Radii (AO3 – analysis)

Connections to Other Syllabus Topics (AO3)

• Thermodynamics (12‑14): Effective temperature is defined by equating a star’s total radiative power to that of a black‑body at temperature T.
• Quantum Physics (22): Derivation of σ from Planck’s law reinforces the quantum origin of radiation.
• Nuclear Physics (23 – optional): Luminosity ultimately comes from fusion; knowing r allows estimation of surface gravity (g = GM/r²) and pressure gradients.
• Mathematical Skills (AO2): Handling powers of ten, square‑roots, and error propagation are explicitly assessed.

Common Pitfalls (AO3 – evaluation)

• Using T instead of T⁴ – error of many orders of magnitude.
• Mixing units (e.g., keeping L in solar units while using σ in SI). Always convert to SI before substitution.
• Forgetting the square‑root when solving for r.
• Neglecting emissivity for stars where ε < 1 (e.g., very cool giants with strong molecular absorption).
• Ignoring uncertainties – temperature errors dominate the radius uncertainty.

Suggested Diagram

Place a Hertzsprung–Russell diagram on the board and overlay curves of constant radius derived from

\$ L = 4\pi\sigma r^{2}T^{4}\quad\Longrightarrow\quad \log L = 2\log r + 4\log T + \log(4\pi\sigma) \$

These curves help visualise why hot, luminous stars can have relatively modest radii, whereas cool supergiants occupy the large‑radius region.

Further Reading / Extensions (optional)

• Derivation of σ from fundamental constants (k, h, c).
• Energy generation pathways in stars – proton–proton chain vs. CNO cycle.
• Observational techniques:

  • Parallax and apparent magnitude → absolute magnitude → luminosity.
  • Spectral classification → effective temperature.

• Radiative transfer in stellar atmospheres – how line blanketing reduces ε.