Answer:
[tex]\frac{AP_{B}}{AP_{S}} = 5.5* 10^4[/tex]
Explanation:
Basically, what the question wants to know is the ratio of Betelgeuse's apparent brightness to that of Sirius.
Apparent brightness (AB) is defined as a function of luminosity (L) and distance (d) as follows:
[tex]AP=\frac{L}{4\pi d^2}[/tex]
Thefore, the ratio between both apparent brightnesses is:
[tex]\frac{AP_{B}}{AP_{S}}=\frac{\frac{8.0*10^9}{4\pi 643^2}}{\frac{26}{4\pi 8.6^2}} \\\frac{AP_{B}}{AP_{S}}=\frac{8.0*10^9*8.6^2}{26*643^2} \\\frac{AP_{B}}{AP_{S}}=55,041.67\\\frac{AP_{B}}{AP_{S}} = 5.5* 10^4[/tex]
Betelgeuse supernova would be 5.5* 10^4 brighter in our sky than Sirius if it were to explode
The luminosity of Betelgeuse is [tex]5.5 \times 10 ^4[/tex] times brighter than that of Sirius
According to the given question, we are to find the ratio of the Betelgeuse to that of Sirius.
Note that the ratio is dependent on their apparent brightness.
[tex]Ratio=\frac{AP_b}{AP_s}[/tex]
Get the apparent brightness of Betelgeuse
[tex]AP_b=\frac{L}{4 \pi d^2} \\AP_b=\frac{8.0\times 10^9}{4 \pi \times 643^2} \\[/tex]
Similarly for apparent brightness of Sirius:
[tex]AP_s=\frac{L}{4 \pi d^2} \\AP_s=\frac{26.0}{4 \pi \times 8.6^2} \\[/tex]
Taking their ratio:
[tex]Ratio =\frac{\frac{8.0\times 10^9}{4 \pi \times 643^2} \\}{\frac{26}{4 \pi \times 8.6^2} \\}[/tex]
On dividing both expressions, the ratio of the apparent brightness of Betelgeuse to that of Sirius is [tex]5.5 \times 10 ^4[/tex]
[tex]\frac{AP_b}{AP_s} = 5.5 \times 10^4\\AP_b = 5.5\times 10^4 AP_S[/tex]
This shows that the luminosity of Betelgeuse is [tex]5.5 \times 10 ^4[/tex] times brighter than that of Sirius.
Learn more here: https://brainly.com/question/3796978
