Answer:
Polaris is approximately 4.796 times farther away than Sirius.
Explanation:
According to the inverse-square law, widely used in wave phenomena, the luminosity, a form of intensity of electromagnetic webs, is inversely proportional to the square of distance. In other words, we get the following definition:
[tex]I \propto \frac{1}{r^{2}}[/tex]
[tex]I = \frac{k}{r^{2}}[/tex] (Eq. 1)
Where:
[tex]k[/tex] - Proportionality ratio, measured in watts.
[tex]r[/tex] - Distance from the center of the star, measured in meters.
[tex]I[/tex] - Luminosity of the star, measured in watts per square meter.
Then, we eliminate the proportionality ratio by constructing the following relationship:
[tex]I_{S}\cdot r_{S}^{2} = I_{P}\cdot r_{P}^{2}[/tex] (Eq. 2)
Where:
[tex]I_{S}[/tex], [tex]I_{P}[/tex] - Intensities of Sirius and Polaris, measured in watts per square meter.
[tex]r_{S}[/tex], [tex]r_{P}[/tex] - Distances from centers of Sirius and Polaris, measured in meters.
After some algebraic handling, we get that:
[tex]\frac{r_{P}}{r_{S}} = \sqrt{\frac{I_{S}}{I_{P}} }[/tex]
If we know that [tex]\frac{I_{S}}{I_{P}} = 23[/tex], then the distance ratio of Polaris to Sirius is:
[tex]\frac{r_{P}}{r_{S}} =\sqrt{23}[/tex]
[tex]\frac{r_{P}}{r_{S}} \approx 4.796[/tex]
In a nutshell, Polaris is approximately 4.796 times farther away than Sirius.
