Stellar Radii (Copy)

1. Wien’s Displacement Law

• Wien’s Law relates the temperature of a star to the wavelength at which it emits the maximum intensity of radiation (λₘₐₓ):

  λₘₐₓ ∝ 1 / T
  or
  λₘₐₓ = b / T

• Where:
  • λₘₐₓ = wavelength of peak emission (m)
  • T = surface temperature of the star (K)
  • b = Wien’s constant ≈ 2.90 × 10⁻³ m·K
• This equation tells us:
  • Hotter stars emit peak radiation at shorter wavelengths (towards the blue end of the spectrum)
  • Cooler stars emit peak radiation at longer wavelengths (towards the red end)

2. Stefan–Boltzmann Law

• Describes the total energy luminosity (L) emitted per second by a black body (e.g. a star) in terms of its temperature and radius:

  L = 4πσr²T⁴

• Where:
  • L = luminosity of the star (W)
  • r = radius of the star (m)
  • T = surface temperature of the star (K)
  • σ = Stefan–Boltzmann constant ≈ 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴
  • 4πr² = surface area of the star assuming it is spherical
• This law implies:
  • Luminosity increases rapidly with temperature (T⁴ relationship)
  • Two stars with the same temperature can have different luminosities due to differences in radius

3. Estimating the Radius of a Star

To find the radius r of a star when luminosity L and temperature T are known:

• Rearranging the Stefan–Boltzmann law:

  r = √(L / (4πσT⁴))

• This allows astronomers to determine stellar radii from observable quantities:
  • Measure L through flux intensity and distance
  • Estimate T using Wien’s displacement law from spectral data

4. Combined Application of Wien’s and Stefan–Boltzmann Laws

• Step 1:
  Use Wien’s law to determine T from the observed peak wavelength λₘₐₓ:

  T = b / λₘₐₓ

• Step 2:
  Use observed luminosity (L) and computed T in the rearranged Stefan–Boltzmann equation to calculate the radius (r).
• This technique is used for stars where distance is known, and spectrum and intensity are measurable.

5. Key Points for Exam

• Always convert λₘₐₓ to meters before using Wien’s Law.
• Stefan–Boltzmann Law assumes a perfect black body — real stars approximate this behavior.
• These laws together are powerful tools in stellar astrophysics:
  • Estimating stellar temperature
  • Finding stellar radii
  • Differentiating between stars of similar temperatures but different sizes and luminosities

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course