Question

Dave is sitting on a spaceship orbiting the Earth (1 au from the sun). At this distance, his solar panels gather an intensity of 1000 (watts/m^2) from the Sun's radiation. Dave wants to gather enough energy to run his coffee machine which needs an intensity of 20 (watts/m^2)

Question 1:

Dave flies to Saturn (which is approximately 10 au from the sun) and at somewhere along the way his coffee machine no longer gets enough power. What intensity do his solar panels collect around Saturn (in (watts/m^2)?

Question 2 (Continuing from 1):

Dave turns around and begins flying back toward the Sun to turn get his coffee maker enough power to turn back on.  At what distance would the coffee maker turn back on (in au)?

Answer 1

Distance of Earth from the Sun = R₁ = 1 au = 1.496 x 10¹¹ m

Intensity gathered by the panels while orbiting Earth = I₁ = 1000 W/m²

Distance of Saturn from the Sun = R₂ = 10 au = 1.496 x 10¹² m

Intensity gathered by the panels around Saturn = I₂

Intensity required for coffee maker to run = I₃ = 20 W/m²

Distance at which the coffee maker would run = R₃

Power emitted by the Sun = P

P = I₁(4R₁²) = I₂(4R₂²) = I₃(4R₃²)

I₁(4R₁²) = I₂(4R₂²)

I₂ = I₁(R₁²/R₂²)

I₂ = 1000[(1.496x10¹¹)²/(1.496x10¹²)²]

I₂ = 10 W/m²

I₁(4R₁²) = I₃(4R₃²)

R₃² = (I₁/I₃)(R₁²)

R₃² = (1000/20)(R₁²)

R₃² = 50R₁²

R₃ = 7.071R₁

R₃ = 7.071 au

1) Intensity his solar panels coleect around Saturn = 10 W/m²

2) Distance at which the coffee maker would turn back on = 7.071 au