Question

How can you calculate solar luminosity using Earth’s solar constant?

Answer 1

Hint: Luminosity is the total electromagnetic radiation of the sun and Earth’s solar constant is the energy received per unit area. The luminosity is related to Earth’s solar constant as well as area. Substituting the corresponding values in the relation, we can calculate luminosity.
Formulas used:
$L=4\pi {{r}^{2}}I$

Complete step-by-step solution:
Luminosity is the total energy of electromagnetic radiation emitted by an object. Since it is energy, it is calculated in watts ($W$).
Earth’s solar constant is the total electromagnetic radiation received by the Earth’s surface per unit area. It includes all types of radiation received from the sun and not just visible light. Its value is $1361\,W{{m}^{-1}}$.
Solar luminosity is calculated from the Earth’s solar constant as-
$\begin{align}
  & L=AI \\
 & \Rightarrow L=4\pi {{r}^{2}}I \\
\end{align}$
Here, $L$ is solar luminosity
$A$ is the area in which the solar radiation spread and $r$ is the distance between sun and Earth
$I$ is Earth’s solar constant
The distance between sun and Earth is $r=1.5\times {{10}^{11}}m$
In the above equation, we substitute given values to get,
$\begin{align}
  & L=4\pi \times {{(1.5\times {{10}^{11}})}^{2}}\times 1361 \\
 & \Rightarrow L=3.83\times {{10}^{26}}W \\
\end{align}$
Therefore, the luminosity of the sun as calculated by Earth’s solar constant is $3.83\times {{10}^{26}}W$.

Additional information: As we go closer to the sun, the value of $r$ decreases and hence luminosity also decreases. The radiation received per unit area of the Earth is also called the solar flux density. The electromagnetic rays received must be perpendicular to the surface area.

Note: The Sun’s radiation is spread in a sphere around it. One part of the earth receives light while the other part is hidden. We can also calculate luminosity using the Stefan Boltzmann law which says $L=\sigma A{{T}^{4}}$ (here, $\sigma $ is Stefan-Boltzmann constant and $T$ is the temperature). It is independent of the earth-sun distance.