Question

At what distance would you have to place the sun so that it would have the same surface brightness

(perceived by the eye) as that of the full moon? Evaluate the reasonableness of your answer and state the assumptions

that you have used in this calculation.

Answer 1

Sun's apparent magnitude is : -26.74

Moon's apparent magnitude is : -12.74

Difference = $x=m_{1}-m_{2}=(-12.74)-(-26.74)=14.00$

Brightness factor : $v_{b}=10^{0.4x}=10^{0.4\times 14.00}\approx 398107.17$ , So the sun's intensity is 400000 times brighter than full moon.

As brightness is inversely proportional to the square of the distance, so we can say that to have the same apparent intensity as the Moon, Sun has to be $\sqrt{398107.17} = 630.9\approx 631$ times further than the Earth-Moon distance, which is 376786km.

So the desired distance of the sun = 376786 * 631 = 2.37*10⁸ km.

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Our Assumptions:

• Earth is still stable when Sun is that far
• All orbits are circular
• Moon is at peak magnitude
• Atmosphere absorption is same for both sunlight and moonlight
• We are considering a perfect eye.