Hi All,
I have a question about Luminosity I'm hoping to get some help with.
Q is: stars of Deneb, and Mimosa, "appear to be the same brightness", with Deneb at 3230 light years away, and Mimosa at 353 light years. Which star has the greater luminosity, and by how much?
Clearly Deneb's intrinsic brightness or luminosity would be stronger, if they appear similarly bright, despite Deneb's much further distance. The second part of the Q is the area we're unsure of.
An example compares the luminosity of Sirius, at a distance of 8.61 light years, to that of the sun. It states Sirius's apparent brightness as 8.8x10^-11. So in a worked example, it coverts 8.61 to AU (x 63 240), to determine the solar distance ratio (R/R⊙), = 5.44x10^5. The question states "we know the brightness ratio, b/b⊙" which is 8.8x10^-11". They then work it all out: L/L⊙ = b/b⊙ x (R/R⊙)^2 =8.8x10^-11 x (5.44x10^5)² = 26. This makes sense.
However, my query is, how exactly did they determine Sirius's apparent brightness?
In previous pages, info is given on apparent magnitude scales (i.e. the sun at -26.7), which prompted us to find Deneb's (+1.25) and Mimosa's (+1.25), there is mention but no example in the book, of the logarithmic scale it's all based on. The book also says "the brightness ratio can be worked out from the apparent magnitudes, but that is a little messy so we shall just quote the brightness ratio directly"
There's another worked example of the sun's luminosity, which uses it's watts per square metre on earth (1370), to calculate it's luminosity: L = b x 4πR² (b as brightness in watts, and 4πR² being the area of a sphere, R as the distance in AU (1) yet converted to metres: 1.5x10^11), it's also described as b = L/4πR².
The page also states the inverse law, or brightness declining as radiation increases with distance, written as a ratio of y₁/y₂ = (x₂/x₁)², or propto of y ∝ 1/x².
Hence, do we use the inverse law to compare the stars? if so, which values do we substitute for x and y (some online examples use AU, most American resources quote in parsec), from the question we're not sure all or the correct info is given, or is it again just a matter of conversions?
Is it best to use the calculation in the Sirius example to determine each stars' brightness ratio to the sun, then compare them to each other? (bearing in mind no watts radiation values are given, and internet searches yield varied results, but most tend to state info like "Deneb is 200,000x brighter than the sun", which I think is only the calculation result.)
Thus, which calculations are used to find each stars' luminosity, or brightness, especially if the equation with Sirius, is used?
Sorry again for all the info, just wasn't sure where exactly to look or which steps to use when (the question's driving me nuts, and has been for 6+ hours!)
Thanks so much again!