Sirius B. The brightest star in the sky is Sirius, the Dog...

Sirius B. The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is $24,000 \mathrm{K}$ and that it radiates energy at a total rate of $1.0 \times 10^{25} \mathrm{W}$ . Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius B? (b) What is the peak-intensity wavelength'? Is this wavelength visible to humans? (c) What is the radius of Sirius $B ?$ Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more total energy per second, the hot Sirius $\mathrm{B}$ or the (relatively) cool sun with a surface temperature of 5800 $\mathrm{K} ?$ To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

Key Concepts

Stefan-Boltzmann Law

This law states that the total energy radiated per unit surface area of a blackbody per unit time (also known as the radiated intensity) is directly proportional to the fourth power of the blackbody's absolute temperature. It is given by I = ?T?, where ? is the Stefan-Boltzmann constant. This relationship is used to determine the radiative intensity and to compute a star's total luminosity when combined with its surface area.

Comparative Stellar Radiation

In astrophysics, comparing the radiated power or luminosity of different stars involves both their temperatures and radii. Because the total radiated energy depends on both the fourth power of the temperature and the square of the radius, even a hotter star with a smaller radius can radiate less energy overall than a larger, cooler star. This comparison helps in understanding the dominance of a star in its energy output relative to others.

Luminosity and Surface Area Calculation

The total power emitted by a star (its luminosity) can be determined by multiplying the radiative intensity by the star’s surface area. For a spherical star, this gives L = 4?R²?T?. This formula is central when computing unknown properties such as the star’s radius from its total power and temperature.

Blackbody Radiation

This concept describes an idealized physical body that absorbs all incident electromagnetic radiation and re-emits energy with a characteristic spectrum that only depends on its temperature. In astrophysical contexts, many stars approximate blackbodies, enabling the use of blackbody radiation laws to relate temperature, emitted energy, and spectral distribution.

Wien's Displacement Law

Wien's Displacement Law relates the temperature of a blackbody to the wavelength at which its emission spectrum is maximized. The law is expressed as ?_max * T = constant. This principle allows one to calculate the peak intensity wavelength, which is critical for determining whether the radiation lies in the visible range or not.

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