Solid Angles of Planets and Which Planet to Use to Build a Dyson Sphere?

Time for something completely different from my usual electronics projects. This morning I started to wonder (don’t ask why) a couple of fundamental questions:

  • If you’d convert the planet Mercury to a huge solar panel, would it cover the sun as viewed from earth?
  • How large portion of sun’s energy you’d get for the project if you started by covering the surface of Mercury with solar panels?
  • Assuming hyper advanced technology that could do this, would Venus or some other planet in our solar system perhaps be a better target?

The thought experiment initiated from some Science podcast, probably from Anatomy of Next scientist interviews (or BBC’s Tomorrow’s World), which talked about turning Mercury into solar panels or mirrors — essentially a Dyson Swarm.

Mercury is close to sun, so it’s a clear candidate, but I started thinking “If you were an alien (or future human race) with super advanced technology arriving to solar system, which planet would you pick?”. You could of course harvest gas giants for Helium-3 and use fusion energy to do stuff, or have some not-yet-invented energy source. But let’s focus on the giant fusion reactor in the center of our solar system, and look at some numbers.

Basic Data for Planets

If we think of sun as a perfect sphere putting out energy in form of solar radiation to pretty much uniformly to all directions, the amount of radiation reaching a certain planet is a simple function of the planet’s radius and it’s distance from sun (π*r^2 / d). Surely you can quickly google which planet gets the most radiation? Turns out most articles on the internet for laymen focus on energy per unit of area (square feet or meters), and leave out the size of the planet completely. Well, easy to fix! Let’s start with some basic data from NASA’s planetary factsheet:

Planet Mass (10^24kg) Diameter (km) Distance from Sun (10^6 km)
Mercury 0.33 4,879 58
Venus 4.87 12,104 108
Earth 5.97 12,756 150
Mars 0.642 6,792 228
Jupiter 1898 142,984 779
Saturn 568 120,536 1,434
Uranus 86.8 51,118 2,873
Neptune 102 49,528 4,495
Pluto 0.0146 2,370 5,906

How Big Do Planets Look From The Sun?

Now as all planets are really far away from the sun compared to their radius and diameter, I’m just doing a shortcut and calculating their angle from sun as just the ratio of diameter and distance, owing to the fact that sin(x) ≈ x for very small values of x (second coefficient of Taylor series expansion is x^3/9 so with these numbers we’ll get about 20 significant digits correct). Calculating diameter/distance shows that even the largest planet from the sun, Jupiter, is just 184 microradians, that is 0.000184 radians wide when viewed from the sun! Full sky would be 3.14 radians, so there’s some way to go!
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