21.1 Luminosity of a Star

21.1 Luminosity of a Star

Luminosity is the total amount of energy emitted by a star per unit time in the form of electromagnetic radiations. It is a fundamental property that defines the intrinsic brightness of a star. Luminosity is represented by the symbol $L$ and is measured in watts, indicating the total power of radiation emitted by a star. Luminosity of a star primarily depends on its radius and surface temperature. Larger stars have more surface area to emit radiations and therefore, tend to be more luminous. Similarly, for stars of equal size, the one with the higher surface temperature will radiate more energy and hence, exhibits greater luminosity. The luminosity of a star can be represented as the product of its surface area ($A=4\pi r^{2}$) and radiant intensity ($I = \sigma T^{4}$).

Equation (21.1) is called Stefan-Boltzmann law. It states that;

The total output power of a star is proportional to the product of its surface area and the fourth power of its temperature. where $\sigma$ is Stefan-Boltzmann constant. Its value is $5.67 \times 10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$, $T$ is the surface temperature and $r$ is the radius of the star.

(Fig. 21.1: The Sun)

Tidbit!

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The Sun shines with an immense luminosity of about $4\times 10^{26}$ watts that is equivalent to billions of nuclear explosions happening every second.

Do you know?

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Stefan-Boltzmann's law helps astronomers estimate the size and brightness of stars just by knowing their temperature.

Example 21.1

The luminosity of a star is given as $2.5 \times 10^{26}\,\mathrm{W}$, and its surface temperature is $6000\,\mathrm{K}$. Estimate the radius of the star.

Solution

Luminosity of the star $L = 2.5 \times 10^{26}\,\mathrm{W}$ Surface temperature of the star $T = 6000\,\mathrm{K}$ Radius of the star $r = ?$ Using Stefan-Boltzmann law; $L = 4\pi r^{2} \sigma T^{4}$ or $r = \sqrt{L / (4\pi\sigma T^{4})}$ $= \sqrt{2.5 \times 10^{26}\,\mathrm{W} / (4 \times 3.14 \times 5.67 \times 10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}} \times (6000\mathrm{K})^{4})}$ $r = 5.2 \times 10^{8}\,\mathrm{m}$

Brain Teaser

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If a star becomes twice as hot, how many times more energy will it radiate?