Two stars, both of which behave like ideal black bodies,...

Two stars, both of which behave like ideal black bodies, radiate the same total energy per second. The cooler one has a surface temperature $T$ and a diameter 3.0 times that of the hotter star. (a) What is the temperature of the hotter star in terms of $T ?$ (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

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Key Concepts

Stefan-Boltzmann Law

This law states that the total power radiated per unit area of a black body is proportional to the fourth power of its temperature. In the context of stars, it is used to relate the luminosity (total energy emission) of a star to its surface temperature and surface area, allowing comparison between stars of different sizes and temperatures when their luminosities are equal.

Surface Area Scaling for Spherical Bodies

For spherical objects like stars, the surface area is proportional to the square of the radius (or diameter). This scaling relationship is crucial when comparing the total energy outputs of stars with different diameters, as a larger star will have a greater radiating area contributing to its overall luminosity.

Wien's Displacement Law

This law relates the temperature of a black body to the wavelength at which its emission peaks, stating that the product of the peak wavelength and the temperature is constant. It is used to compare the peak-intensity wavelengths of stars with different temperatures, illustrating how hotter stars emit most strongly at shorter wavelengths compared to cooler stars.

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