How Do You Determine a Star's Radius from Luminosity and Temperature?

For both SL and HL IB Physics, determine a star's radius by substituting its luminosity and absolute surface temperature into the Stefan-Boltzmann law and rearranging:

$$R=\sqrt{\frac{L}{4\pi\sigma T^4}}$$

Here, $L$ is luminosity, $T$ is absolute surface temperature, and $\sigma$ is the Stefan-Boltzmann constant.

The Reasoning

A star is approximated as a spherical black body. Its luminosity is the total power radiated across its surface and is related to surface area and effective surface temperature by:

$$L=\sigma AT^4$$

For a spherical star, the surface area is $A=4\pi R^2$. Substitution gives:

$$L=4\pi R^2\sigma T^4$$

Divide by $4\pi\sigma T^4$, then take the positive square root because radius is a magnitude:

$$R^2=\frac{L}{4\pi\sigma T^4}$$ $$R=\sqrt{\frac{L}{4\pi\sigma T^4}}$$

This relationship also shows how radius scales with the measured quantities: $R\propto L^{1/2}T^{-2}$. Therefore, doubling luminosity at constant temperature increases radius by a factor of $\sqrt{2}$, while doubling temperature at constant luminosity reduces radius by a factor of four. This proportional reasoning is useful for comparing stars without completing a full calculation.

Use $\sigma=5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$, provided in the IB Physics data booklet. Before calculating, confirm that luminosity is total power and that temperature is in kelvin.

Worked Example

A star has $L=3.83\times10^{26}\ \text{W}$ and $T=5800\ \text{K}$. Therefore:

$$R=\sqrt{\frac{3.83\times10^{26}}{4\pi(5.67\times10^{-8})(5800)^4}}$$ $$R=6.9\times10^8\ \text{m}$$

This is approximately the Sun's radius. The fourth-power dependence means temperature strongly affects the result: for fixed luminosity, a hotter star must have a smaller radius.

A common misconception is substituting apparent brightness for luminosity. Apparent brightness is power received per unit area and depends on distance; luminosity is the star's total power output. If only apparent brightness $b$ and distance $d$ are given, first calculate $L=4\pi d^2b$. Temperature must always be converted to kelvin.

Exam Technique

This is assessed in E.5 Fusion and stars for SL and HL. For Calculate, show the equation, substitution, and answer. For Show, display every algebraic step, including $A=4\pi R^2$. Check that the final radius is in metres and has a physically reasonable order of magnitude.