Let's break down the two parts of this problem:

(a) Stefan-Boltzmann Law Problem

We are given two stars: the Sun and another star. We need to compare their luminosities using Stefan-Boltzmann's Law, which states:

$L = 4\pi R^2 \sigma T^4$

Where:

• $L$ is luminosity,
• $R$ is the radius of the star,
• $T$ is the surface temperature, and
• $\sigma$ is the Stefan-Boltzmann constant.

The question asks to find how many times more or less luminous the second star is compared to the Sun. To solve this, we need to compute the ratio of luminosities:

$\frac{L_{\text{star}}}{L_{\odot}} = \frac{4 \pi R_{\text{star}}^2 \sigma T_{\text{star}}^4}{4 \pi R_{\odot}^2 \sigma T_{\odot}^4} = \left(\frac{R_{\text{star}}}{R_{\odot}}\right)^2 \left(\frac{T_{\text{star}}}{T_{\odot}}\right)^4$

We are given:

• $R_{\odot} = 1 R_{\odot}$, $T_{\odot} = 5800 \, \text{K}$,
• $R_{\text{star}} = 2 R_{\odot}$, $T_{\text{star}} = 5000 \, \text{K}$.

Substituting these into the equation: