(j) luminosity L of a star; Stefan’s law L = 4πr²σT⁴ where σ is the Stefan…

The idea

Luminosity L is the total electromagnetic power a star radiates in all directions, measured in watts (W). Stefan’s law models a star as a black-body sphere: L = 4πr²σT⁴, where r is the star’s radius in metres, T is its surface temperature in kelvin, and σ is the Stefan constant, about 5.67 × 10⁻⁸ W m⁻² K⁻⁴. The 4πr² factor is the star’s surface area, so L scales with surface area and with the fourth power of temperature. Doubling r multiplies L by 2² = 4; doubling T multiplies L by 2⁴ = 16. Rearranged, r = √(L / (4πσT⁴)). Check units: W m⁻² K⁻⁴ × m² × K⁴ gives W.

What earns the marks

• Luminosity L is the total radiant power emitted by a star in all directions, measured in watts (W).
• Stefan’s law states L = 4πr²σT⁴, where r is stellar radius in metres and T is surface temperature in kelvin.
• The term 4πr² is the surface area of the spherical star, so L is proportional to r² and to T⁴.
• σ is the Stefan constant, approximately 5.67 × 10⁻⁸ W m⁻² K⁻⁴, and is a universal constant.
• Rearranging gives r = √(L / (4πσT⁴)), allowing radius to be found when L and T are known.
• Unit checking confirms W m⁻² K⁻⁴ × m² × K⁴ = W, consistent with luminosity in watts.

Common mistakes and how to put them right

• Treating L as brightness received at Earth rather than total power emitted.

  Put it right: L is an intrinsic property of the star, while observed intensity depends on distance.

• Using T in degrees Celsius instead of kelvin.

  Put it right: Convert by adding 273 to the Celsius value before raising T to the fourth power.

• Forgetting to square the radius or raising T to the wrong power.

  Put it right: Apply L = 4πr²σT⁴ exactly, with r² and T⁴.

• Using the diameter in place of the radius.

  Put it right: Halve the diameter first, since the formula uses r.

Exam tips

• Write the rearranged form r = √(L / (4πσT⁴)) before substituting numbers to reduce algebra slips.
• Convert temperatures to kelvin and radii to metres before substituting into the equation.
• Sanity-check the answer: a hotter star of the same radius must have a much larger luminosity because of the T⁴ dependence.