engineering and design

Steradians (the original essay)

Steradians are a measure of solid angles. At one time I couldn't find anything on the web about them, now there's plenty. Examples: Field of Vision and Astronomy. It would also be in calculations of how much energy radiates between two bodies, what fraction of the complete "sphere" an object is surrounded by is occupied by another object that it is radiating to, or receiving radiation from.

It's amazing to me that the surface area of a sphere is exactly 4 times the area of a circle going through its middle. So the area is 4 pi r^2. A radian is the angle of radius r rolled on a circle. A steradian is the solid angle of area r^2 rolled onto a sphere. So 4 pi steradians is the solid angle of a sphere, about 12 steradians. 2 pi steradians is the solid angle of a hemisphere, about 6 steradians. I guess a human field of vision is about 5 steradians, we don't see a full hemisphere do we? So about 1/5 of what we see is an angle of 1 steradian.

How many steradians does the full moon occupy? Say the diameter of the moon is 2159 miles, so its flat area to our vision is about 3,661,000 square miles. Say the distance of the moon to the earth is 238854 miles, so the surface area of a sphere centered at earth and intersecting the moon is about 4 pi 238854^2 = 716,900,000,000 square miles. Now, 3,661,000 / 716,900,000,000 = 0.000 005 1. So the moon is occupying 0.000 005 1 of the whole sphere. The whole sphere is 4 pi steradians, so 0.000 005 1 times 4 pi is 0.000 064, so the full moon occupies about 0.000 064 steradians when viewed from the earth. Not much.

How about my hand? It's about an average of 6 inches by 4 inches for 24 square inches. When I hold it out in front of me its about 26 inches from my eyes. The surface area of a sphere centered at my eyes with 26 inch radius is 4 pi 26^2 = 8495 square inches. So my hand occupies 24 / 8495 = 0.002 825 of that sphere, or 0.002 825 times 4 pi = 0.035 5 steradians. Still not much.

Check. Ratio of steradians: Hand/ Moon = 0.0355 / 0.000064 = 555. Would it take 555 full moons to fill the area of my hand held out in front of me? Well, I happen to have a full moon available at the moment. Hmmm. The full moon is about the size, or maybe only half the area of my pinky nail. That's a surprise, but that's all it is. So I guess it would take a few hundred moon areas to fill up my hand area, so our "check" indicates this and all of the above is about right.