Space-VURD

[triadruid]

On a completely different scale/note from my last post, a thought-experiment has been rattling around my brain, about what would happen if you scaled down a planet's diameter but kept the same mass/gravity. That led to discovering this fine gem, which is ostensibly a "how much energy would it take the Death Star to blow up your constructed world" calculator, but also gives numbers for things like planetary diameter at different densities and the like. Unless something's wrong with the calculator, however, I don't understand why holding surface gravity constant at 1g while decreasing diameter would result in a decreasing mass, as well as increasing density. What am I doing wrong here? It seems to calculate fine for the inner terrestrials, but if I decrease diameter to 10,000km and hold gravity steady, mass decreases by almost 40%!

That seems impossible, unless I missed something in elementary physics... ::edit:: and of course, I did: Newton's Law of Universal Gravitation, and I screwed up gravity and gravitation. Thanks liquidfun!

Comments

[fionnabhar.livejournal.com]

Unless something's wrong with the calculator, however, I don't understand why holding surface gravity constant at 1g while decreasing diameter would result in a decreasing mass, as well as increasing density.

This is the part where the sparkly silver woo-woo fairy flits in, waves her magic wand, wiggles her fingers, and throws some glitter. And poof, there you go, decreased mass and increased destiny. Or, erm, density.

It's something like that, anyway.

[liquidfun.livejournal.com]

Your average distance to the mass of the planet is decreasing, and gravitational attraction is proportional to the inverse square of the distance. Because you are holding gravitationl force constant at 1g and decreasing the distance, the mass required goes down quickly ...

Assuming you can simplify the mass of a body to a point mass at the center of the body, we can use some simple equations...

Given relationship:
Force(grav) = Mass(body1) * Mass(body2) / Distance^2

Substituting what we have:
1g = PlanetaryMass * ReferenceMass / PlanetaryRadius^2

Solving for PlanetaryMass:
PlanetaryMass = 1g * PlanetaryRadius^2 / ReferenceMass

So if we have two different PlanetaryRadii, one twice as large as the other, the mass PlanetaryMass required for the smaller Planet is only 1/4 of that of the larger one, from an idealized point of view ... it's actually not quite that extreme, due to not really being able to assume a point mass at the center of the body, but you get the idea ...

Yay, physics. Not as much fun as magic, but it is still fun.

[fionnabhar.livejournal.com]

That's what I said just not in those words. Duh. *slaps forehead*

[liquidfun.livejournal.com]

I bow to your sparkly wand-waving.

*bow*

[fionnabhar.livejournal.com]

Oh, not mine. I'm just explaining. I live in a glitter-free zone. :-)

[liquidfun.livejournal.com]

Yeah, the "=" should really be a symbol indicating proportionality unless we throw in a gravitational constant but I don't think we're trying for amazing accuracy here ...

[triadruid.livejournal.com]

That's awesome that you were able to pull that up from memory so quickly. I had sorted out that surface gravity was not the same as gravitation after a while, and was just coming back here to correct myself... what I forgot, of course, was Newton, *headdesk*

Now, to sort out how atmospherics is affected by smaller diameter, and a higher iron content at the core... or maybe I'll do some work.

[fionnabhar.livejournal.com]

Gravity, gravitation. Destiny, density. Easy mistake to make, really.

[opaljax.livejournal.com]

You are the cutest geek, really, almost as cute as your girlfriend. :D

[teross50.livejournal.com]

Pass the scotch, and let's do some more cool shit, like that!

[featherynscale.livejournal.com]

Unrelated to astrophysics, do you know mhael? He seems to be your evil WoT/otter/poly-guy-with-a-wife-and-girlfriend/star-wars-geeking twin. Also, he stole your Dumai's Wells icon.

[triadruid.livejournal.com]

Huh. He definitely seems vaguely familiar, but I'm not sure if we met him at D*Con or what.