| Syllabus topic | Connection to this lesson |
|---|---|
| 7 Waves – electromagnetic radiation | Wien’s law describes the wavelength at which a black‑body spectrum peaks; the relation c = λ f explains how spectrographs record λ. |
| 5 Work, Energy & Power | Stefan–Boltzmann law gives the radiant power per unit area (W m⁻²); multiplying by the star’s surface area yields the total power (luminosity). |
| 14 Temperature | Effective temperature is obtained from the peak wavelength; it is the temperature a perfect black‑body would need to emit the observed total power. |
| 12 Gravitation (A‑level) | Once R is known, surface gravity follows from g = GM / R², allowing a link to the mass‑radius relationship. |
| 25 Astronomy & Cosmology | The trio (L, R, T) places a star on the Hertzsprung–Russell diagram (main‑sequence, giant, super‑giant). |
| 22 Quantum Physics (A‑level) | Black‑body radiation originates from Planck’s law; Wien’s law is the high‑temperature approximation used here. |
| Constant | Symbol | Value | Units |
|---|---|---|---|
| Wien’s constant | b | 2.898 × 10⁻³ | m·K |
| Stefan–Boltzmann constant | σ | 5.670374419 × 10⁻⁸ | W·m⁻²·K⁻⁴ |
| Solar luminosity | L⊙ | 3.828 × 10²⁶ | W |
| Solar radius | R⊙ | 6.96 × 10⁸ | m |
| Solar absolute bolometric magnitude | Mbol,⊙ | 4.74 | mag |
Electromagnetic radiation obeys c = λ f, where c is the speed of light, λ the wavelength and f the frequency. A spectrograph disperses the incoming light so that a detector records flux as a function of λ, producing the spectral‑energy distribution (SED) used in this unit.
For a perfect black‑body the wavelength at which the spectral radiance per unit wavelength is maximum is
\[
\lambda_{\max}= \frac{b}{T}
\]
where b = 2.898 × 10⁻³ m·K. A brief derivation (high‑temperature limit of Planck’s law) can be shown in a few steps:
Thus hotter stars have a shorter λmax.
The total radiant emittance (power per unit area) from a black‑body surface is
\[
j^{\star}=σT^{4}\qquad\bigl[\text{W·m}^{-2}\bigr]
\]
Multiplying by the surface area of a sphere gives the total luminosity:
\[
L = 4πR^{2}σT^{4}
\]
The temperature obtained from Wien’s law is the effective temperature – the temperature a perfect black‑body would need to emit the same total power as the star. It is not necessarily the local temperature at any point on the stellar surface, but the assumption of a uniform temperature is required for the simple \(L=4πR^{2}σT^{4}\) relation.
When the stellar mass M is known, the surface gravity follows from Newton’s law:
\[
g = \frac{GM}{R^{2}}
\]
where G = 6.674 × 10⁻¹¹ N m² kg⁻². This provides a bridge to the A‑level gravitation topic.
Planck’s law (1900) solved the ultraviolet catastrophe and underpins the black‑body spectrum. Wien’s law is the high‑temperature (short‑wavelength) approximation of Planck’s law, which is why it is appropriate for most hot stars.
If the measured peak wavelength and luminosity have uncertainties \(\Delta\lambda\) and \(\Delta L\), the fractional errors are
\[
\frac{\Delta T}{T}= \frac{\Delta\lambda}{\lambda_{\max}},\qquad
\frac{\Delta R}{R}= \frac12\left(\frac{\Delta L}{L}+4\frac{\Delta T}{T}\right)
\]
Consequences:
\[
T = \frac{b}{\lambda_{\max}},\qquad
\Delta T = T\frac{\Delta\lambda}{\lambda_{\max}}
\]
\[
d\;(\text{pc}) = \frac{1000}{\pi}
\]
\[
M = m - 5\log_{10}(d) + 5
\]
\[
M_{\text{bol}} = M + \text{BC}
\]
\[
\frac{L}{L{\odot}} = 10^{\;(M{\text{bol},\odot}-M_{\text{bol}})/2.5}
\]
where \(M_{\text{bol},\odot}=4.74\).
\[
L = \left(\frac{L}{L{\odot}}\right)L{\odot}
\]
Propagate uncertainties from distance (parallax error) and photometric error to obtain \(\Delta L\).
\[
R = \sqrt{\frac{L}{4\pi\sigma T^{4}}},\qquad
\frac{\Delta R}{R}= \frac12\left(\frac{\Delta L}{L}+4\frac{\Delta T}{T}\right)
\]
Express the final answer in solar radii:
\[
\frac{R}{R_{\odot}} = \frac{R\;(\text{m})}{6.96\times10^{8}\ \text{m}}
\]
\[
g = \frac{GM}{R^{2}}
\]
Goal: Using a real stellar spectrum, determine the star’s radius and write a concise evaluation of the result.
Follow the steps in §3‑4 to obtain \(L\), \(\Delta L\) and finally \(R\) and \(\Delta R\).
| Aspect | Comments / Evidence |
|---|---|
| Measurement uncertainties (λ, m, π) | |
| Assumption of black‑body behaviour | |
| Interstellar extinction (estimate E(B–V) if possible) | |
| Bolometric correction accuracy | |
| Possible systematic errors (limb darkening, instrumental response) | |
| Suggested improvement (e.g., interferometry, eclipsing binary analysis) |
Write a short (≈150 words) paragraph addressing the points above – this satisfies AO3.
\[
T_{\odot}= \frac{2.898\times10^{-3}}{5.00\times10^{-7}} = 5.80\times10^{3}\ \text{K},
\qquad
\frac{\Delta T}{T}=1\% \Rightarrow \Delta T\approx58\ \text{K}
\]
\[
R_{\odot}= \sqrt{\frac{3.828\times10^{26}}{4\pi(5.670374419\times10^{-8})(5.80\times10^{3})^{4}}}
= 6.96\times10^{8}\ \text{m}
\]
\[
\frac{\Delta R}{R}= \frac12\bigl(0+4\times0.01\bigr)=2\%
\Rightarrow \Delta R\approx1.4\times10^{7}\ \text{m}
\]
Given:
Calculations:
\[
T = \frac{2.898\times10^{-3}}{1.0\times10^{-6}} = 2.90\times10^{3}\ \text{K},
\qquad
\frac{\Delta T}{T}=5\%
\]
\[
R = \sqrt{\frac{1.2\times10^{5}\,L_{\odot}}{4\pi\sigma(2.90\times10^{3})^{4}}}
= 8.6\times10^{11}\ \text{m}
\]
\[
\frac{\Delta R}{R}= \frac12\bigl(0.05+4\times0.05\bigr)=0.125\;(12.5\%)
\]
\[
R = (8.6\pm1.1)\times10^{11}\ \text{m}
= (1\,240\pm160)\ R_{\odot}
\]
The huge radius places Betelgeuse firmly in the red‑supergiant region of the HR diagram.
Stellar classification & the HR diagram
The three observables (L, R, T) locate a star on the Hertzsprung–Russell diagram. Main‑sequence stars follow roughly \(L\propto M^{3.5}\) and have relatively small radii for their luminosities. Giants and super‑giants lie above the main sequence because their radii are vastly larger at comparable temperatures.
Black‑body radiation in the early universe
The Cosmic Microwave Background is a near‑perfect black‑body at 2.73 K. Using the Stefan–Boltzmann law one can calculate its energy density, linking this lesson to the quantum‑physics requirement (Planck’s law).
Other methods of measuring stellar radii
• Interferometry (direct angular diameter → physical radius with distance).
• Eclipsing binaries (light‑curve modelling gives relative radii).
• Asteroseismology (oscillation frequencies constrain mean density → radius).
Discussing these methods reinforces AO3 – evaluation of experimental techniques.
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