Stellar radii
Section: Astronomy and Cosmology | Syllabus: Cambridge A Level Physics (9702)
Stellar Radii
Wien's Displacement Law
All objects emit electromagnetic radiation. The wavelength at which the radiation intensity is maximum depends on the object's temperature.
Wien's Displacement Law The peak wavelength of radiation emitted by a black body is inversely proportional to its absolute temperature. _max ∝ (1/T)or: _max T = constant = 2.90 × 10^-3 m K Key Implications
- Hotter objects emit radiation at shorter wavelengths (appear bluer)
- Cooler objects emit radiation at longer wavelengths (appear redder)
- The Sun (T ≈ 5800 K) has peak emission in the visible range (yellow)
Graphs of intensity vs wavelength for objects at different temperatures. Hotter objects have curves that peak at shorter wavelengths and have greater total area (higher luminosity).
The Stefan-Boltzmann Law
The luminosity of a star depends on both its surface temperature and its size.
L = 4π σ r^2 T^4where L = luminosity (W), σ = Stefan-Boltzmann constant (5.67 × 10^-8 W m^-2 K^-4), r = radius of star (m), T = surface temperature (K) Key Points
- Luminosity depends on r^2 - doubling the radius increases luminosity by a factor of 4
- Luminosity depends on T^4 - doubling the temperature increases luminosity by a factor of 16
- 4π r^2 is the surface area of the star
Estimating the Radius of a Star
By combining Wien's displacement law and the Stefan-Boltzmann law, we can estimate the radius of a star.
Method- Measure the peak wavelength _max from the star's spectrum
- Use Wien's law to calculate the surface temperature T
- Measure or estimate the luminosity L
- Rearrange Stefan-Boltzmann law to find radius r
r = √(L/4π σ T^4)
| Star Type | Typical Radius | Example |
|---|---|---|
| Supergiant | 100-1000 R_ | Betelgeuse |
| Giant | 10-100 R_ | Aldebaran |
| Main sequence | 0.1-10 R_ | Sun, Sirius |
| White dwarf | ~0.01 R_ | Sirius B |
| Neutron star | ~10 km | Pulsars |
Worked Examples
Worked Example: Surface Temperature from Peak WavelengthQuestion: The Sun has a surface temperature of 5780 K and peak emission at 480 nm. Another star has peak emission at 250 nm. Estimate its surface temperature.
SolutionUsing _max T = constant:
_1 T_1 = _2 T_2
T_2 = (_1 T_1/_2) = (480 × 5780/250) = 11\,100 K
Worked Example: Calculating LuminosityQuestion: Calculate the luminosity of the Sun given: radius = 6.96 × 10^8 m, surface temperature = 5780 K.
SolutionL = 4π σ r^2 T^4
L = 4π × 5.67 × 10^-8 × (6.96 × 10^8)^2 × (5780)^4
L = 4π × 5.67 × 10^-8 × 4.84 × 10^17 × 1.12 × 10^15
L = 3.85 × 10^26 W
Worked Example: Estimating Stellar RadiusQuestion: The star Sirius A has a surface temperature of 10,000 K and luminosity of 9.9 × 10^27 W. Estimate its radius.
Solutionr = √(L/4π σ T^4)
r = √9.9 × 10^274π × 5.67 × 10^-8 × (10\,000)^4
r = √9.9 × 10^277.13 × 10^9
r = √1.39 × 10^18 = 1.2 × 10^9 m
This is about 1.7 times the radius of the Sun.
Worked Example: Using Both Laws TogetherQuestion: A black body at 2000 K has peak emission at 1450 nm. The star Betelgeuse has peak emission at 850 nm and luminosity 3.1 × 10^31 W. Estimate its radius.
SolutionStep 1: Find temperature using Wien's law:
T = (2000 × 1450/850) = 3410 K
Step 2: Calculate radius:
r = √3.1 × 10^314π × 5.67 × 10^-8 × (3410)^4
r = √3.1 × 10^319.64 × 10^7 = √3.2 × 10^23
r = 5.7 × 10^11 m (about 820 times the Sun's radius)
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