Smaller Planets

[Cheiromancer]

If the radius were half that of earth, the density of the planet would have to be twice that of earth to have the same gravitational pull. About 11 times that of water, IIRC. The earth is largely iron, I think, so twice that density.

The distance of the horizon would be 70% what it is now; the square root of one half.

 

[Delta]

One more thing: distance from the sun... this planet is smaller, so I guess the planet would have to be farther? Does that have any effects?

No, it doesn't.

 

[Nonlethal Force]

But who does it matter to?

It doesn't have to matter to anyone except the OP. I imagine that the OP is a bit like myself in that if they DM they like to understand the mechanics of the world that they are creating. There is nothing wrong with rule zero and just making it work ... but in this case the OP is asking for a discussion on physical mechanics - and probably only for his/her own understanding. As a fellow DM, I appluad the effort to understand the effects of world generation rather than defaulting to rule zero! Especially since it is likely that none of this will come up in game.

Actually, the fact that none of this will come up in game is probably a sign that they world is well built and well thought out. As a player I get irritated when DMs have to rely too heavily on "it's a magical world, deal with it" or "your characters would have grown up in this world so it is their understanding of normal." That can be fun for a while and it is indeed quite true, but I also like to be able to not have to suspend reality and scientific side once and a while. I enjoy the academic pursuit of what effects changing things like world density and radius would have! As I said, I applaud the OP's effort for the fact that it will likely never come up in game if the world is well thought out.

I like a good purely academic question once and a while.

If the radius were half that of earth, the density of the planet would have to be twice that of earth to have the same gravitational pull. About 11 times that of water, IIRC. The earth is largely iron, I think, so twice that density.

Is this right? I don't know the formula's connecting density and gravity, so I'm not saying that it is wrong. I'm just curious. I know that mass is often a function of volume and volume is a function of radius cubed in a spherical world. But, I also know that gravity is a linear force of attraction between two masses. Just curious, that's all.

 

[Cheiromancer]

A planet with half the radius will have 1/8 the volume (volume is proportional to radius cubed). Gravity increases at a rate inversely proportional to the square of the distance between the two bodies. Since the radius is halved, you are going to get 4 times the gravity per unit mass. But if the density is the same, the mass will be only 1/8 as much, resulting in 4 * 1/8 = 1/2 as much gravity. So you have to increase the density by a factor of 2 to compensate.

I seem to recall that the distance to the horizon is the square root of the product of the height of your eyes off the ground with the diameter of the planet you are on. Does that sound right?

 

[Kae'Yoss]

You're the DM, you can kick that LawsOfNature's pansy ass if you want to. Nothing changes unless you tell it to change.

The only thing that would change would be that planetars would have less HD!

 

[BRP2]

Okay, so far, it seems pretty easy to understand. I'll just make the core something more mythical, like a Adamantium core.

The distance of the horizon would be 70% what it is now; the square root of one half.

I don't get what this means entirely.

 

[Nonlethal Force]

Nothing terribly drastic. It means that if you go out onto a flat field here on the earth and the distance from you to the furthest point you can see on the earth is 20 miles then the distance you can see to the horizon on a planet with half the radius in 70% of that, or 14 miles.

NOTE: I've no idea what the distance to the horizon actually is, I just made up a number that make the math purposes easier.

 

[Delta]

Distance to horizon derivation: http://www-istp.gsfc.nasa.gov/stargaze/Shorizon.htm
Assume 6 foot man (0.0018 km, as used below).

D = sqrt(2Rh).
D sight distance in kilometers.
R radius of planet in kilometers.
h height of eye in kilometers.

* For Earth, radius R = 6378 km.
So, sight distance D = sqrt(2 * 6378 * 0.0018) = 4.79 km.

* For Mini-Earth, radius R = 3189 (half size).
So, sight distance D = sqrt(2 * 3189 * 0.0018) = 3.39 km.

Note the latter is about 70% the former: 3.39/4.79 = 0.71.
Same as sqrt(1/2) = 0.71.

 

[helium3]

A lower mass planet orbiting around our sun at the same distance as Earth and with the same atmospheric composition would be cooler (perhaps significantly) than our own world. The lower mass of planet means a thinner atmosphere, which means that it doesn't trap heat as well as ours. To compensate, you'd need to move the planet closer to the sun or assume that there's a higher proportion of "greenhouse" gases in the atmosphere to compensate. One solution would result in a shorter "year" than our own and the other would maybe cause a change in the color of the sky or the types of lifeforms that are best suited to the environment. You could deal with the shortened year problem by increasing the rate at which the planet rotates. Essentially, their day is objectively shorter than ours but it's not like anyone notices since there's no other reference point.

Or you could assume that the core consists of molten adamantium or some other imaginary hyperdense material.

Lower gravity isn't such a big deal, since again, it's not like anyone native to that world knows anything is wrong. A lower gravity world would objectively have a greater amount of vertical relief between features, since the driving force for smoothing everything else out isn't quite as strong as it is on our world.

 

[Arkhandus]

To the OP:

Basically, a smaller planet just results in 1) lower gravity and 2) a much more noticeable curve to the horizon and land, as perceived by creatures living there. That's pretty much it. #1 can be solved by saying that the planet's core is proportionately larger and denser than Earth's (taking up a larger percentage of the planet's volume), and that means it's probably composed of molten adamantine or something. I forget if gold, platinum, or copper is heavier than iron, off the top of my head...... If so, then they might be better sources for the planet's molten core. #2 can probably be solved by saying that there composition of the atmosphere is a bit different, and obscures vision (or distorts the visible curvature of the land) moreso than Earth's over long distances, to the eyes of normal creatures.

Regarding one poster's mention of a differently-colored sky: he has a bit of a point there, all of this means pretty much nothing to the PCs, usually. Also: to note, the human eye doesn't see the spectrum of light in quite the proper 'purity' or whatever......since we developed on a world with blue skies and a yellow sun, we perceive colors as having a bit of a yellowed hue, compared to the true spectrum of light, IIRC. So what we perceive as being white is actually a bit yellow, and what we perceive as being red is actually a bit orange, or something like that. So in a setting where creatures are born under the natural developments of a world with orange skies, they would not perceive it as being anything odd or unusual. But if they were transported to an alternate Material Plane that resembled our own, they'd see everything being unusual in coloration; a wall that we painted white may look, I dunno, fuschia or gray or dark green or something, to an inhabitant from that orange-skied world (moreso if their own sun wasn't a yellow star). This is all postulation, mind you; I'm no scientist and cannot say with authority how exactly it would work, but this is my understanding.