Question

b) The distance of the red supergiant Betelgeuse is approximately 427 light years. If it were to explode as a supernova, it would be one of the brightest stars in the sky. Right now, the brightest star in the sky other than the Sun is Sirius (which has a luminosity of 26LSun and is 26 light years away). How much brighter than Sirius would the Betelgeuse supernova be (from our point of view) if it reached a maximum luminosity of 10^10LSun? c) There have been some claims that when Betelgeuse explodes it will be like having a second Sun in the sky. Compare Betelgeuse’s brightness to the Sun’s brightness at Earth. Is this likely to be correct?

Answer 1

b) Betelgeuse would be
`\approx 1.43 \cdot 10^(6)` times brighter than Sirius

c) Since Betelgeuse brightness from Earth compared to the Sun is
`\approx 1.37 \cdot 10^(-5) }` the statement saying that it would be like a second Sun is incorrect

Step-by-step explanation:

The start brightness is related to it luminosity thought the following equation:

`B = \displaystyle{(L)/(4\pi d^2)}` (1)

where
`B` is the brightness,
`L` is the star luminosity and
`d`, the distance from the star to the point where the brightness is calculated (measured). Thus:

b)
`B_(Betelgeuse) = \displaystyle{(10^(10)L_(Sun))/(4\pi (427\ ly)^2)}` and
`B_(Sirius) = \displaystyle{(26L_(Sun))/(4\pi (26\ ly)^2)}` where
`L_(Sun)` is the Sun luminosity (
`3.9 x 10^(26) W`) but we don't need to know this value for solving the problem.
`ly` is light years.

Finding the ratio between the two brightness we get:

`\displaystyle{(B_(Betelgeuse))/(B_(Sirius))=(10^(10)L_(Sun))/(4\pi (427\ ly)^2) * (4\pi (26\ ly)^2)/(26L_(Sun)) \approx 1.43 \cdot 10^(6) }`

c) we can do the same as in b) but we need to know the distance from the Sun to the Earth, which is
`1.581 \cdot 10^(-5)\ ly`. Then

`\displaystyle{(B_(Betelgeuse))/(B_(Sun))=(10^(10)L_(Sun))/(4\pi (427\ ly)^2) * (4\pi (1.581\cdot 10^(-5)\ ly)^2)/(1\ L_(Sun)) \approx 1.37 \cdot 10^(-5) }`

Notice that since the star luminosities are given with respect to the Sun luminosity we don't need to use any value a simple states the Sun luminosity as the unit, i.e 1. From this result, it is clear that when Betelgeuse explodes it won't be like having a second Sun, it brightness will be 5 orders of magnitude smaller that our Sun brightness.