Stellar radii
Section: Astronomy and Cosmology | Syllabus: Cambridge AS 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.
λ_max 1/T or: λ_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) Figure: Black Body Curves 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^4 where 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 λ_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 From L = 4π σ r^2 T^4, rearranging for 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 Wavelength Question: 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. Solution Using λ_max T = constant: λ_1 T_1 = λ_2 T_2 T_2 = λ_1 T_1/λ_2 = 480 × 5780/250 = 11\,100 K Worked Example: Calculating Luminosity Question: Calculate the luminosity of the Sun given: radius = 6.96 × 10^8 m, surface temperature = 5780 K.
Solution L = 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 Radius Question: The star Sirius A has a surface temperature of 10,000 K and luminosity of 9.9 × 10^27 W.
Estimate its radius. Solution r = √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 Together Question: 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.
Solution Step 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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