How many AUs do you need to be from the Sun for the brightness to be 3 times bigger?

Question

You can use the module show to yourself that the Earth receives 1300 Watts per square meter when it is 1 AU from the Sun.

How many AUs do you need to be from the Sun for the brightness to be 3 times bigger?

Answer 1

The apparent brightness declines with the square of your distance: ie, if you increase your distance by a factor of 2, the brightness declines by a factor of 2 squared (4).

For an absolute brightness 'X' (which is a constant), at distance 'R(1)', where R(1) = 1 AU, the apparent brightness 'B(1)' = X/(R(1)^2).

Rearrange: (R(1)^2)B(1) = X

You want to triple the brightness: new brightness we'll call B(2):

(R(2)^2)B(2) = X. (Because 'X', the absolute brightness, is a constant regardless of distance).

Therefore: (R(2)^2)B(2) = (R(1)^2)B(1)

B(2) = 3B(1) (because you have tripled the brightness).

Substitute:

3(R(2)^2)B(1) = (R(1)^2)B(1)

B(1) cancels:

3(R(2)^2) = R(1)^2



Rearrange: R(1) = sq rt[3(R(2)^2)] = 3^(1/2)R(2) = R(2)

Therefore, R(2) = 1/[3^(1/2)]R(1)

R(1) = 1 AU, therefore, R(2) = 1/[3^(1/2)]AU ('one divided by root 3' AU, about 0.577 AU).

Answer 2

1/4 au.

Answer 3

I'd guess 0.333~ AU, but I'm probably wrong.