Question

Sirius is 22 times more luminous that the Sun, and Polaris is 2350 times more luminous that the Sun. Sirius appears 23 times brighter than Polaris. Polaris is ______ times farther away than Sirius.

Answer 1

Polaris is approximately 4.796 times farther away than Sirius.

Step-by-step 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:

`I \propto (1)/(r^(2))`

`I = (k)/(r^(2))` (Eq. 1)

Where:

`k` - Proportionality ratio, measured in watts.

`r` - Distance from the center of the star, measured in meters.

`I` - Luminosity of the star, measured in watts per square meter.

Then, we eliminate the proportionality ratio by constructing the following relationship:

`I_(S)\cdot r_(S)^(2) = I_(P)\cdot r_(P)^(2)` (Eq. 2)

Where:

`I_(S)`,
`I_(P)` - Intensities of Sirius and Polaris, measured in watts per square meter.

`r_(S)`,
`r_(P)` - Distances from centers of Sirius and Polaris, measured in meters.

After some algebraic handling, we get that:

`(r_(P))/(r_(S)) = \sqrt{(I_(S))/(I_(P)) }`

If we know that
`(I_(S))/(I_(P)) = 23`, then the distance ratio of Polaris to Sirius is:

`(r_(P))/(r_(S)) =√(23)`

`(r_(P))/(r_(S)) \approx 4.796`

In a nutshell, Polaris is approximately 4.796 times farther away than Sirius.