recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the peak surface temperature of a star

Stellar Radii

Learning Objective

Recall and use Wien’s displacement law

\$\lambda_{\max} = \frac{b}{T}\$

(where \$b \approx 2.898 \times 10^{-3}\,\text{m·K}\$) to estimate the peak surface temperature of a star from its observed peak wavelength.

Key Concepts

• Black‑body radiation: Stars approximate black‑bodies, emitting a continuous spectrum that peaks at a wavelength inversely proportional to their surface temperature.
• Wien’s displacement law: Relates the wavelength of maximum emission (\$\lambda_{\max}\$) to temperature (\$T\$). The law can be written as

  \$\lambda_{\max} \propto \frac{1}{T}\$

  or in its full form shown above.

• Peak wavelength measurement: Determined from the star’s spectrum (e.g., using a spectroscope or photometric filters).
• Temperature estimation: Rearranging Wien’s law gives

  \$T = \frac{b}{\lambda_{\max}}\$

  allowing a quick temperature estimate.

Step‑by‑Step Procedure

1. Obtain the star’s spectrum and identify the wavelength at which the intensity is maximum (\$\lambda_{\max}\$). Units should be metres (m) or nanometres (nm).
2. Convert \$\lambda_{\max}\$ to metres if necessary (1 nm = \$10^{-9}\$ m).
3. Insert \$\lambda_{\max}\$ into Wien’s law:

   \$T = \frac{2.898 \times 10^{-3}\ \text{m·K}}{\lambda_{\max}}\$

4. Calculate \$T\$ and express the result in kelvin (K).
5. If required, compare the temperature with known stellar classifications (e.g., O, B, A, … M).

Worked Example

Suppose a star’s spectrum shows a peak at \$\lambda_{\max}=500\,\text{nm}\$.

1. Convert to metres: \$500\,\text{nm}=5.0\times10^{-7}\,\text{m}\$.
2. Apply Wien’s law:

   \$T = \frac{2.898\times10^{-3}}{5.0\times10^{-7}} \approx 5.80\times10^{3}\ \text{K}\$

3. The estimated surface temperature is \$\approx 5800\ \text{K}\$, comparable to the Sun’s effective temperature.

Typical Peak Wavelengths and Corresponding Temperatures

Common Pitfalls

• Forgetting to convert nanometres to metres before using the constant \$b\$.
• Assuming the observed peak wavelength is not affected by interstellar reddening; in practice, extinction can shift the apparent \$\lambda_{\max}\$ to longer wavelengths.
• Using the wrong value for the Wien constant; \$b = 2.898\times10^{-3}\,\text{m·K}\$ is the standard value.

Extension: Relating Temperature to Stellar Radius

Once the temperature \$T\$ is known, the star’s luminosity \$L\$ can be related to its radius \$R\$ via the Stefan‑Boltzmann law:

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

where \$\sigma = 5.670\times10^{-8}\,\text{W·m}^{-2}\text{K}^{-4}\$.

Rearranging gives the radius:

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

Thus, temperature estimates from Wien’s law are a crucial first step in determining stellar radii.