(455) Stellar radius [IB Physics SL/HL]

Finding the Radius of a Star

Key Concepts

• Black Body Behavior: Stars can be treated as black bodies in physics, allowing the use of laws related to thermal radiation.

Essential Laws

• Wien's Displacement Law: Relates the peak wavelength (BB_max) to temperature (T).

  • Formula:[ \lambda_{\text{max}} \cdot T = 2.90 \times 10^{-3} \text{ m K} ]

• Stefan-Boltzmann Law: Relates luminosity (L) to temperature and surface area:

  • Formula:[ L = \sigma \cdot A \cdot T^4 ]

  • Where:

    • L = Luminosity (in watts)

    • A = Surface area (for a sphere A = 4\u03C0R²)

    • T = Surface temperature (in Kelvin)

    • (\sigma = 5.67 \times 10^{-8} \text{ W m}^{-2} \text{K}^{-4})

Finding Temperature

• Using Wien's Law to determine the surface temperature of a star:

  • Example with Sirius A (peak wavelength: 291 nm):

  • Calculation:[ T = \frac{2.90 \times 10^{-3}}{291 \times 10^{-9}} \approx 9970 \text{ K} ]

Finding Radius

• Use Stefan-Boltzmann Law to derive the radius of a star:

  • Rearranged formula:[ R = \sqrt{\frac{L}{4 \pi \sigma T^4}} ]

• Example with Sirius A (L = 25.4 L_\odot, L_\odot = 3.85 \times 10^{26} W):

  • Luminosity calculated:[ L \approx 9.85 \times 10^{27} \text{ W} ]

  • Calculation for R: [ R \approx 1.18 \times 10^{9} \text{ m} \approx 1.18 \text{ million km} ]

HR Diagram Context

• Position of Sirius A on HR Diagram:

  • Surface Temperature: ~9000 K

  • Luminosity: ~25 times that of the Sun

  • Radius: ~1.7 times that of the Sun

• Comparison with the Sun and other stars on the diagram:

  • Demonstrates relative scale and characteristics of Sirius A compared to the Sun.