A temperature from two wavelengths

Adapted from IOAA 2014 (Suceava), theory problem P7. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

The light of a star is measured in two narrow bands of the same width, at wavelengths ${\lambda }_1$ and ${\lambda }_2$, and the flux for each unit of wavelength is $I_1$ in the first and $I_2$ in the second. Treat the star as a blackbody obeying the Planck law.

(a) Write down the equation that gives the star's temperature $T$ from these measurements.

(b) Find $T$ explicitly when $hc/\lambda kT$ is much more than 1 at both wavelengths.

(c) When $hc/\lambda kT$ is much less than 1 at both wavelengths and $I_1=2I_2$, how are ${\lambda }_1$ and ${\lambda }_2$ related?

AnswerThe general equation is the ratio of two Planck curves. When $hc/\lambda kT$ is large, $T=hc({\lambda }_1-{\lambda }_2)/(k{\lambda }_1{\lambda }_2\ln [(I_1/I_2)({\lambda }_1/{\lambda }_2{)}^5])$, and when it is small, $I_1=2I_2$ requires ${\lambda }_2=2^{1/4}{\lambda }_1\approx 1.19{\lambda }_1$.