Estimating the Radius of Stars (OCR A Level Physics): Revision Note

Estimating the Radius of Stars

• The radius of a star can be estimated by combining Wien’s displacement law and the Stefan–Boltzmann law

• The procedure for this is as follows:

  • Use Wien’s displacement law to find the surface temperature $T$ of the star

  • Use the inverse square law of intensity equation to find the luminosity $L$ of the star (if given the intensity $I$ and stellar distance $d$)

  • Then, use the Stefan-Boltzmann law to determine the radius $r$ of the star

Summary of equations

Inverse square law of intensity

• Starting from the equation for the intensity of a progressive wave:

$I=\frac{P}{A}$

• For a star:

  • the power output is its luminosity, so $P=L$

  • the area over which the light spreads is $A=4\mathrm{\pi }{d}^2$

• Therefore, the inverse square law of intensity for a star is:

$I=\frac{L}{4\mathrm{\pi }{d}^2}$

• Where:

  • $I$ = intensity of light received on Earth (W m^-2)

  • $L$ = luminosity of the star (W)

  • $d$ = distance between the star and the Earth (m)

Wien's displacement law

• Wien's law for a star is given by:

${\lambda }_{max}T=\mathrm{constant}$

• Where:

  • ${\lambda }_{max}$ = wavelength emitted by the star at maximum intensity (m) 

  • $T$ = surface temperature of the star (K) 

Stefan-Boltzmann law

• Stefan's law for a star is given by:

$L=4\mathrm{\pi }{r}^2\sigma T^4$

• Where:

  • $L$ = luminosity of the star (W)

  • $r$ = radius of the star

  • $\sigma$ = the Stefan-Boltzmann constant

  • $T$ = surface temperature of the star (K)