23.2 Luminosity and Radiant Flux Intensity

Luminosity

Luminosity ($L$) is the total electromagnetic energy emitted by a star per unit time — it is the star's total power output.

$L=\text{total power radiated by the star}$

SI unit: Watts ($\mathrm{W}$) or $\mathrm{J}{\mathrm{s}}^{-1}$

Luminosity is an intrinsic property of a star — it does not depend on how far away the observer is. It depends on the star's surface temperature and surface area.

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Stefan-Boltzmann Law and Luminosity

The Stefan-Boltzmann Law states that the total power radiated per unit surface area of a black body is proportional to the fourth power of its absolute temperature:

$E=\sigma T^4$

where $\sigma =5.67\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ is the Stefan-Boltzmann constant.

For a spherical star of radius $r$ and surface temperature $T$, the total surface area is $4\pi r^2$, so the luminosity is:

$\boxed{L=4\pi r^2\sigma T^4}$

This shows that luminosity depends on two factors:

1. The surface area of the star (related to its radius $r$)
2. The surface temperature $T$ (raised to the fourth power)

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Radiant Flux Intensity

Radiant Flux Intensity ($F$), also called apparent brightness, is the power received per unit area at a distance $d$ from a star of luminosity $L$.

The star radiates energy equally in all directions. At distance $d$, this energy is spread over a sphere of surface area $4\pi d^2$:

$\boxed{F=\frac{L}{4\pi d^2}}$

SI unit: Watts per square metre ($\mathrm{W}{\mathrm{m}}^{-2}$)

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Inverse Square Law

From the formula $F=\frac{L}{4\pi d^2}$, it is clear that:

$F\propto \frac{1}{d^2}$

This is the Inverse Square Law for radiant flux intensity: if the distance to a star is doubled, the radiant flux intensity decreases by a factor of 4.

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Worked Example

Problem: The Sun has luminosity $L_{\odot }=3.85\times 10^{26}\mathrm{W}$ and is at a distance of $1.5\times 10^{11}\mathrm{m}$ from Earth. Calculate the radiant flux intensity at Earth's surface.

Solution:

$F=\frac{L}{4\pi d^2}=\frac{3.85\times 10^{26}}{4\pi (1.5\times 10^{11}{)}^2}$

$F=\frac{3.85\times 10^{26}}{4\pi \times 2.25\times 10^{22}}=\frac{3.85\times 10^{26}}{2.83\times 10^{23}}$

$F\approx 1360\mathrm{W}{\mathrm{m}}^{-2}$

This value (~$1360\mathrm{W}{\mathrm{m}}^{-2}$) is known as the solar constant.

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Summary Table