## **23.2 LUMINOSITY AND RADIANT FLUX INTENSITY** This section defines and differentiates between luminosity and radiant flux intensity, two fundamental concepts in astronomy related to electromagnetic radiation emitted by celestial objects. It introduces the Stefan-Boltzmann Law and the Inverse Square Law as key principles governing these quantities. ### **Luminosity (L)** * The total amount of electromagnetic energy emitted per unit time by a star or other celestial object. * Represents the total *power output* of radiation from a star. * **SI Units**: Joules per second ($\mathrm{J} \mathrm{s}^{-1}$) or Watts ($\mathrm{W}$). * **Dependence**: Luminosity depends on the *temperature* and *surface area* of the star. ### **Stefan-Boltzmann Law** The Stefan-Boltzmann Law states that the total energy emitted by a blackbody per unit area per second is proportional to the fourth power of its absolute temperature. * For a blackbody, luminosity (L) is given by: $$ \mathrm{L}=\mathrm{A \sigma T}^{4} $$ Where: * $A$ is the surface area. * $T$ is the absolute temperature (in kelvins). * $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$). * For a spherical star with radius $r$, the surface area $A = 4\pi r^2$. Thus, $$ \mathrm{L}=4 \pi r^{2} \sigma \mathrm{T}^{4} $$ * *Implication*: If two stars have the same temperature, the star with double the radius will have four times the luminosity. ### **Radiant Flux Intensity (F)** * The luminosity per unit area measured at a specific distance from a light source (e.g., on Earth). * Describes how bright a celestial object appears from a given distance. * Light from a star spreads uniformly through a spherical shell as it travels. The area of this shell is $4\pi d^2$, where $d$ is the distance from the star. * **SI Units**: Watts per square meter ($\mathrm{W} \mathrm{m}^{-2}$). ### **Relationship with Luminosity** $$ F = \frac{L}{A} = \frac{L}{4 \pi d^{2}} $$ Where: * $L$ is the luminosity of the source. * $d$ is the distance from the source. ### **Inverse Square Law** Radiant flux intensity follows an inverse square law, meaning $F \propto 1/d^2$. If the distance doubles, the intensity decreases by a factor of four. This is similar to the behavior of fields discussed in . * **Distinction**: Luminosity (L) is the total power output of the star itself, while radiant flux intensity (F) is what is measured on Earth and depends on the distance to the star. A greater radiant flux intensity indicates a closer star if luminosities are equal. ### **Standard Candles** In astronomy, a "Standard Candle" is an object of known luminosity. By measuring its radiant flux intensity ($F$) on Earth, astronomers can use the inverse square law formula to calculate the distance ($d$) to the object and its host galaxy. ### **Possible Questions/Answer** - **Q:** A star has a luminosity of $9.7 \times 10^{27} \mathrm{~W}$. If the radiant flux intensity measured on Earth is $114 \mathrm{nW} \mathrm{m}^{-2}$, determine the distance of the star from Earth. **A:** * Given: $L = 9.7 \times 10^{27} \mathrm{~W}$, $F = 114 \mathrm{nW} \mathrm{m}^{-2} = 114 \times 10^{-9} \mathrm{~W} \mathrm{~m}^{-2}$. * Formula: $F = \frac{L}{4 \pi d^{2}}$ * Rearranging for $d$: $d = \sqrt{\frac{L}{4 \pi F}}$ * Calculation: $d = \sqrt{\frac{9.7 \times 10^{27}}{4 \pi (114 \times 10^{-9})}} = 8.2 \times 10^{16} \mathrm{~m}$. - **Q:** What is the relative luminosity of objects A and B if $T_A = 100 \mathrm{~K}$, $T_B = 200 \mathrm{~K}$, $R_A = 10 \mathrm{~m}$, and $R_B = 5 \mathrm{~m}$? **A:** * Formula for luminosity: $L = 4 \pi r^2 \sigma T^4$. * Ratio of luminosities: $\frac{L_A}{L_B} = \frac{4 \pi R_A^2 \sigma T_A^4}{4 \pi R_B^2 \sigma T_B^4} = \left(\frac{R_A}{R_B}\right)^2 \left(\frac{T_A}{T_B}\right)^4$ * Substitute values: $\frac{L_A}{L_B} = \left(\frac{10}{5}\right)^2 \left(\frac{100}{200}\right)^4 = (2)^2 \left(\frac{1}{2}\right)^4 = 4 \times \frac{1}{16} = \frac{1}{4}$ * So, $L_A = \frac{1}{4} L_B$, or $L_B = 4 L_A$. ### **Summary Table** | Quantity | Symbol | Definition | Formula | SI Unit | | :---------------------------- | :----- | :------------------------------------------------------------------------ | :---------------------------------------- | :---------------------------- | | **Luminosity** | L | Total power output of a star | $L = 4\pi r^2 \sigma T^4$ | J/s or W | | **Radiant Flux Intensity** | F | Luminosity per unit area measured at a distance $d$ | $F = \frac{L}{4\pi d^2}$ | $\mathrm{W} \mathrm{m}^{-2}$ | | **Stefan-Boltzmann Constant** | $\sigma$ | Constant relating energy emission to temperature | $5.67 \times 10^{-8}$ | $\mathrm{W m}^{-2} \mathrm{~K}^{-4}$ | Understanding luminosity and radiant flux intensity allows astronomers to determine intrinsic properties of stars (like their energy output) and their distances from Earth, despite how faint they may appear.

23.2 Luminosity And Radiant Fluxintensity | Cosmology | Physics 12