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

Estimating the Surface Temperature of a Star – Wien’s Displacement Law

Learning Objectives (AO1, AO2, AO3)

• AO1 – Knowledge & Understanding: recall Wien’s displacement law, its constant, and the related wave‑particle relations; recognise the symbols and SI units used.
• AO2 – Handling & Evaluating Information: convert between λ, ν and photon energy E; propagate uncertainties; apply significant‑figure rules; assess the linearity of a λ_max vs 1/T plot.
• AO3 – Experimental Skills: plan, carry out and evaluate a practical investigation; record data in a lab notebook (Paper 5); discuss safety and sources of error (Paper 3).

1. Key Formulae and SI Units (AO1)

2. Wave‑Particle Connections (Syllabus links 7 Waves, 22 Quantum physics)

• Wavelength ↔ Frequency: ν = c / λ  [Hz = m s⁻¹ / m]
• Photon Energy: E = hν = hc / λ  [J = J·s · Hz = J·s · m s⁻¹ / m]
• These relations allow the peak of a stellar spectrum to be expressed as λ_max, ν_max or E_max, linking the wave description (7) with the particle description (22).

3. Using Wien’s Law to Find Temperature (AO2)

1. Identify λ_max from the observed spectrum (e.g. the wavelength at maximum intensity).
2. Convert to metres before using the constant b.

   
   1 nm = 1 × 10⁻⁹ m.

3. Calculate the temperature

   \[

   T = \frac{b}{\lambda_{\max}} \qquad\text{(K)}

   \]

4. Uncertainty propagation (first‑order approximation)

   \[

   \frac{\Delta T}{T}= \frac{\Delta\lambda}{\lambda_{\max}}

   \]

   where Δλ is the absolute uncertainty in λ_max.

5. Significant figures:

   • The result for T should be reported with the same number of significant figures as the least‑precise measurement (usually λ_max).
   • Round ΔT to one‑significant figure and then round T to the same decimal place.

4. Data‑Analysis Checklist (AO2)

• ✓ All quantities expressed in SI units before calculation.
• ✓ Record the number of significant figures for each measurement.
• ✓ Propagate uncertainties using the relative‑error formula.
• ✓ Plot λ_max (x‑axis) against 1/T (y‑axis); fit a straight line and note the slope (should equal b).
• ✓ Evaluate the linearity: calculate R², examine residuals, identify outliers.
• ✓ Include error bars on the plot and comment on their size.
• ✓ Complete a brief evaluation of systematic effects (e.g., interstellar reddening, instrument resolution).

5. Worked Example (including significant figures & uncertainty)

Observed peak wavelength: λ_max = 500 ± 5 nm

1. Convert to metres: λ = 5.00 × 10⁻⁷ m  Δλ = 5 × 10⁻⁹ m.
2. Temperature:

   \[

   T = \frac{2.898\times10^{-3}}{5.00\times10^{-7}} = 5.80\times10^{3}\ \text{K}

   \]

   (3 s.f. because λ has 3 s.f.)

3. Relative uncertainty:

   \[

   \frac{\Delta T}{T}= \frac{5.0\times10^{-9}}{5.00\times10^{-7}} = 0.010 = 1.0\%

   \]

   Hence ΔT = 0.010 × 5800 K ≈ 58 K → round to 60 K (1 s.f.).

4. Result (to the same decimal place as ΔT):

   \[

   T = (5.80 \pm 0.06)\times10^{3}\ \text{K}

   \]

   which matches the Sun’s effective temperature (≈ 5800 K).

6. Reference Table – Peak Wavelengths, Frequencies, Photon Energies, Temperatures and Spectral Types

7. Common Pitfalls & How to Avoid Them (AO2)

• Unit conversion errors: always write λ_max in metres before using b.
• Incorrect significant figures: keep track of s.f. at each step; do not give a temperature with more s.f. than the measurement permits.
• Neglecting interstellar reddening: dust can shift the observed peak to longer λ, causing an underestimate of T. Mention this in the evaluation.
• Using a wrong value for b: the accepted value is 2.898 × 10⁻³ m·K (three significant figures).
• Omitting uncertainty propagation: reporting a temperature without ΔT does not satisfy AO3 requirements.

8. Practical Investigation (AO3 – Paper 3 & 5)

Goal: Verify Wien’s displacement law with a laboratory black‑body source and determine the constant b.

Equipment

• Black‑body radiator (e.g., a tungsten filament or a calibrated lamp) with a controllable temperature range 800 K – 1800 K.
• Digital thermocouple or pyrometer (± 10 K) for independent temperature measurement.
• Portable spectrometer or diffraction grating + calibrated CCD/photodiode detector (wavelength resolution ≤ 2 nm).
• Computer with data‑acquisition software (to plot intensity vs. wavelength).
• Safety goggles, heat‑resistant gloves, and a heat‑proof stand.

Safety

• Handle hot filaments with gloves; allow the source to cool before moving it.
• Wear safety goggles when the source is bright or when the spectrometer uses a laser alignment beam.
• Ensure the work area is free of flammable materials.

Procedure

1. Set up the black‑body source on a stable stand and connect the thermocouple.
2. Align the spectrometer so that the detector receives light directly from the source.
3. Heat the source to a chosen temperature (e.g., 800 K). Record the thermocouple reading (T_actual) and note the time.
4. Acquire a spectrum, identify the wavelength of maximum intensity (λ_max) and record its value and the instrument’s wavelength uncertainty (Δλ, typically ± 2 nm).
5. Repeat steps 3–4 for at least two additional temperatures (e.g., 1200 K and 1600 K).
6. For each data set calculate T_Wien = b/λ_max and its uncertainty ΔT using the formula in Section 3.
7. Plot λ_max (x‑axis) against 1/T_actual (y‑axis). Fit a straight line; the slope should be close to b. Record the slope, its standard error, and R².
8. Complete a lab notebook entry (Paper 5) containing:

   • aim, equipment list, raw data tables, calculations, plots with error bars, and a concise evaluation.

Evaluation (Paper 3)

• Discuss random errors (e.g., wavelength resolution, temperature reading fluctuations).
• Identify systematic errors (e.g., spectrometer calibration, interstellar‑like reddening from the glass window, non‑ideal black‑body emissivity).
• Suggest improvements (e.g., use a calibrated black‑body cavity, higher‑resolution spectrometer, apply a correction for emissivity).

9. Linking the Content to the Cambridge 9702 Syllabus

• 7 Waves – Electromagnetic spectrum: the star’s radiation is treated as EM waves; λ_max is a characteristic point on the spectrum.
• 22 Quantum physics – Energy & momentum of a photon: conversion between λ, ν and E demonstrates wave‑particle duality.
• 25 Astronomy & Cosmology (A‑level): temperature is a key axis on the Hertzsprung–Russell diagram; links to stellar classification.
• AO1, AO2, AO3 requirements: the note provides definitions, calculations, uncertainty handling, a practical investigation, safety considerations and an evaluation framework that meet the assessment objectives.

10. Extension – From Temperature to Stellar Radius (A‑level only)

When the effective temperature T is known, the Stefan‑Boltzmann law relates it to the star’s luminosity L and radius R:

\[

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

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

\]

In an A‑level project students can:

1. Obtain the apparent brightness (flux) from photometric data.
2. Use a parallax or other distance measurement to convert flux to luminosity.
3. Insert the temperature derived from Wien’s law into the equation above to calculate the stellar radius.