CAIE A-Level Physics 25.2 Stellar Radii

Practise using Wien’s law and the Stefan-Boltzmann law to find stellar temperature, luminosity or radius and combine them for stellar estimates.

Exam points

use Wien?s displacement law to infer stellar surface temperature from peak wavelength
use the Stefan?Boltzmann law to calculate stellar luminosity, radius or temperature
combine Wien?s law and the Stefan?Boltzmann law to estimate stellar radius

Covered syllabus points

25.2.1—Wien’s displacement law λmax ∝ 1 / T to estimate the peak surface 25.2.2—The Stefan–Boltzmann law L = 4 πσr 2 T 4 25.2.3—Wien’s displacement law and the Stefan–Boltzmann law to estimate the radius

[Maximum number: 3]

(a)

The radius of the Sun is $6.96 \times 10^{8} \mathrm{~m}$ .

Show that the temperature T of the surface of the Sun is 5770 K .

[ 1 ]

$\begin{aligned} \mathrm{L}=4 \pi \sigma r^{2} \mathrm{~T}^{4} 3.83 \times 10^{26}=4 \times \pi \times 5.67 \times 10^{-8} \times 6.96 \times 10^{82} \times \mathrm{T}^{4} \text { leading to } \mathrm{T}=5770 \mathrm{~K} \end{aligned}$

B1

(b)

The wavelength $\lambda_{\text {max }}$ of light for which the maximum rate of emission occurs from the Sun is $5.00 \times 10^{-7} \mathrm{~m}$ .
The temperature of the surface of the star Sirius is 9940 K .
Use information from (d) to determine the wavelength of light for which the maximum rate of emission occurs from Sirius.
wavelength =
m

[ 2 ]

$\begin{aligned} \lambda_{(\max )} \propto \frac{1}{\mathrm{~T}} \frac{5.00 \times 10^{-7}}{\lambda}=\frac{9940}{5770} \end{aligned}$

C1

$\lambda=2.90 \times 10^{-7} \mathrm{~m}$

A1

[Maximum number: 2]

The radiant flux intensity of the radiation from the star in (b) is $2.52 \times 10^{-8} \mathrm{Wm}^{-2}$ when observed at a distance of $4.16 \times 10^{16} \mathrm{~m}$ from the star.

Determine the radius of the star.
radius = m

$\begin{aligned} L=4 \pi \sigma r^{2} T^{4} 5.48 \times 10^{26}=4 \pi \times 5.67 \times 10^{-8} \times r^{2} \times 6980^{4} \end{aligned}$

C1

$r=5.69 \times 10^{8} \mathrm{~m}$

A1

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Covered syllabus points

25.2 Stellar radii question 1

Question (a)

Question (b)

25.2 Stellar radii question 2