Size of earth to solar system vs drop of water to Mt Everest?
April 25, 2014 12:35 PM   Subscribe

Can someone please tell me (if you could show calculations, that would be wonderful) what the ratio of ratios would be for these two numbers: a) the volume of the earth compared to that of the solar system and b) the volume of a drop of water compared to that of Mount Everest?
posted by shivohum to Science & Nature (9 answers total) 3 users marked this as a favorite
Question #1 might be a little vague; the solar system isn't really a fixed volume as such. You could define it as a spheroid of roughly the same diameter as the orbit of the furthest planet. Or, it might be more meaningful to compare just in terms of diameter for that one.
posted by jeffjon at 12:38 PM on April 25, 2014

Volume of Mount Everest is also a little vague. From sea level, I've seen something (can't find backup, sorry), that it's 1.5*10^12 cubic meters, or 1.5*10^15 L. A drop of water is about 1/10 mL, I think, so the ratio there is 1.5*10^19.

jeffjon already pointed out that "volume of the solar system" is poorly defined. Neptune's average distance from the sun is 4,503,000,000 km, for an average solar system diameter somewhere around 9*10^12 meters. Earth's diameter is 1.3*10^7 m or so. (It's not a sphere.) Diameter ratios and volume ratios aren't directly comparable, but there you are.
posted by supercres at 12:46 PM on April 25, 2014

Or in the alternative, do you mean "the volume of all the individual objects in the solar system, added up"? Because that would be way less than a spheroid centered at the sun extending out to Neptune!

Also I have to point out: "Diameter ratios and volume ratios aren't directly comparable, but there you are" - indeed, volumes vary as the cube of diameter! So this comparison is off by many orders of magnitude.
posted by Joey Buttafoucault at 12:51 PM on April 25, 2014 [1 favorite]

The distance from the sun to pluto is about 3.7 billion miles. If we use that as the radius of a sphere, then the volume of that sphere (with the sun at the center) would be about 2x10^29 cubic miles. The volume of earth is about 2.6x10^11 cubic miles. The ratio of the solar system sphere volume to the earth volume is therefore 7.7x10^17.

If we take a drop of water to have a radius of say 2 mm, then the volume would be about 33.5 cubic mm. Mt. Everest is tricky. Most of its volume is actually in the crustal root way below sea level--similar to the way most of an iceberg is below the water line. But measured from sea level, the volume of rock in Mt Everest is about 365 cubic miles. This works out to be about 1.5x10^21 cubic mm. So the ratio of Mt Everest volume to the volume of a drop of water is therefore about 4.5x10^19.

Note the two ratios are different by a factor of about 100. But pretty close as approximations go.
posted by Seymour Zamboni at 1:01 PM on April 25, 2014 [7 favorites]

volumes vary as the cube of diameter
very minor correction: cube of the radius
posted by soelo at 1:08 PM on April 25, 2014

volumes vary as the cube of diameter
very minor correction: cube of the radius

Both statements are true.

And Seymour Zamboni has about as accurate an answer as you're going to get. And a "nice" answer, to boot!
posted by grog at 1:27 PM on April 25, 2014 [1 favorite]

the volume of a sphere is 4/3 x pi x the radius cubed. you can use whatever units you want if, you have cleared them with the pontiff.
posted by bruce at 2:07 PM on April 25, 2014

It's funny that the solar system calculation is actually a lot easier than the Mt. Everest Calculation.

Personally I'd use the distance to Voyager I which is sometimes, depending on who you ask, considered to be at the "edge" of the solar system. Voyager I is 19 billion kilometers away from the sun. That's 1.9 x 10^13 m. The Earth is roughly 6000 km, or 6 x 10^6 m. So the ratio of those cubes is roughly 10^39:10^21. That's 10^18:1 Mount Everest is 10^4 m tall. Assume it's a sphere (kidding, that's a physics joke), but I could assume it's a right cone with a base that's twice its height. So it's volume is 10^10 cubed time pi divided by 3. Let's just call it cubed, so it's 10^12 meters cubed. I'm just going to say that a drop is a milliliter because that's easier, which is 1 cm cubed so it's 10^-6 m cubed. So the ratio of everest to a drop of water is 10^12:10^-3 or 10^18:1. So the ratio of ratios is 10^18:10^18 so that's 1:1. Hey, neat.

I had to rework my math up there a bit, but I think I got it. It would be even neater to compare the volume of all the water on the earth to the volume of the solar system to the water drop and mount everest.
posted by runcibleshaw at 3:49 PM on April 25, 2014

FWIW, I learned (...either in Chem I or in doing darkroom photography, pipetting Rodinal...) that a drop of water is roughly 0.05 CC/mL (~20 drops to the mL). So say a factor of twenty less than a mL.

OTOH, using the orbit of Neptune as the size of the solar system gives a sphere of only 4.5 billion km radius....
posted by AsYouKnow Bob at 7:44 PM on April 25, 2014

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