Two stars A and B, of luminosity 0.5 and 4.5 times the luminosity of the Sun, respectively— are observed to have the same apparent brightness. Which one is more distant, and how much farther away is it than the other?

Sorry I miss one class and dont know how to do this particular question

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• 4 years ago

Since quantum gave you the elegant answer, I'll give you the Rube Goldberg version.

m = apparent magnitude

M = absolute magnitude

d = distance (parsecs)

log = the base 10 logarithm function

m₁ − M₁ = 5 log(d₁) − 5

m₂ − M₂ = 5 log(d₂) − 5

M₁−M₂ = 2.5 log(L₂/L₁)

L₂/L₁ = 4.5 / 0.5 = 9

M₁ = 2.3856063 + M₂

(m₁−2.3856063−M₂) = 5 log(d₁) − 5

(m₂−M₂) = 5 log(d₂) − 5

m₁ = m₂

m₁ = 5 log(d₁) − 5 + 2.3856063 + M₂

m₁ = 5 log(d₂) − 5 + M₂

5 log(d₁) + 2.3856063 = 5 log(d₂)

10^[5 log(d₁) + 2.3856063] = 10^[5 log(d₂)]

243 d₁⁵ = d₂⁵

d₂/d₁ = (243)⁰·² = 3

The intrinsically brighter star is 3 times farther away than the intrinsically dimmer star is.

• 4 years ago

If they appear to have the same brightness, then the dimmer star is closer, while the brighter star is further away.

Light reduces with the square of the distance; in this case, the brighter star is 4.5 / 0.5 = 9 times brighter than the dimmer star. Since they both appear to be the same luminosity, then the brighter star will be sqrt(9) or 3 times further away.