Hollow Planet Mathematics

[Kaodi]

I would not be surprised if someone has explored this topic already, in which case perhaps someone could direct me to that exploration, but I was wondering if anyone had a good idea of how big a hollow planet would need in order for it to at least feel like you were walking straight while walking along the inside curvature of the world?

 

[Hand of Evil]

May find it: Hollow Earth - Wikipedia, the free encyclopedia

But visability is based on atmosphere and the ability to see the sky. Too many clouds, glare and trees could keep the sky from clear sight.

But that said,

At sea level the curvature of the earth limits the range of vision to 2.9 miles. The formula for determining how many miles an individual can see at higher levels is the square root of his altitude times 1.225. Thus on a clear day at 1,000 feet a person with normal vision can see 39 miles; at 10,000 feet, 123 miles; at 25,000 feet, 194 miles. With good visibility a pilot at 25,000 feet can see Germany from the English Channel; at the same altitude over Tunisia he can see the middle of Sicily.

 

[Beginning of the End]

I would not be surprised if someone has explored this topic already, in which case perhaps someone could direct me to that exploration, but I was wondering if anyone had a good idea of how big a hollow planet would need in order for it to at least feel like you were walking straight while walking along the inside curvature of the world?

Depends on what you mean. Barring some weird atmospherics, the fact that the horizon is visibly curving up is basically never going to stop being an issue.

If you're just talking about "the ground beneath my feet feels flat to me", then you're talking about the threshold being of a sufficiently small enough size that you're no longer talking about a "planet". (Check out O'Neill colonies. They have some weird gravity effects because it's being generated by rotation, but the actual curvature is large enough that you wouldn't feel like you were walking on a curved surface.)

To put it another way: The earth is also a curved surface, albeit in the opposite direction. So if your hollow planet is Earth-size, you shouldn't have any problems.

 

[Haltherrion]

If I recall my high school calculus correctly, the gravity inside a hollow sphere is zero. The gravitational pull of the shell cancels where ever you are within the shell, be it the inner surface or some point above the inner surface.

As a thought experiment consider the 2D case of being within a circle. In the center you are equidistant from the circle surface. Move along the +x axis you approach the mass of the +x side of the circle but you have that much more mass "behind you" pulling away.

Similarly if you were at the center of the earth, you would experience zero gravity. Of course you would also experience crushing pressure...

 

[Kaodi]

If you're just talking about "the ground beneath my feet feels flat to me", then you're talking about the threshold being of a sufficiently small enough size that you're no longer talking about a "planet". (Check out O'Neill colonies. They have some weird gravity effects because it's being generated by rotation, but the actual curvature is large enough that you wouldn't feel like you were walking on a curved surface.)

This is more like what I was talking about, though big enough it is more like, " the ground looks reasonably flat to me, " such that at least if you shot an arrow the surface distance that could be covered would be neglibily affected by the curvature.

 

[El Mahdi]

If I recall my high school calculus correctly, the gravity inside a hollow sphere is zero. The gravitational pull of the shell cancels where ever you are within the shell, be it the inner surface or some point above the inner surface...

Yup. The only way to generate a comparable force to gravity would be through rotation of the shell, thus generating an artificial gravity through centrifugal force. But, even in that scenario the "gravity" wouldn't be uniform. The further you move away from the spins equator (toward the spins axis), the less the effect of centrifugal force until it reached zero at the poles. The habitable area inside the sphere/world would be more like a Niven Ring, with most of the remaining interior surface area unuseable. It would be more practical to simply make a ring world in the first place.

 

[Achan hiArusa]

Yes, by applying Guass's Law to gravity, the interior of a spherical shell will have 0-gravitational pull. To get 1 g of spin gravity a planet the size of the Earth would have to spin at the rate of 17 rev/day (set Newton's Second Law for gravity equal to Centripetal Force, mg = mv^2/r). Then the force of gravity would be the cosine of the latitude (1 g at the equator and 0 at the poles), but the force of gravity doesn't necessarily have to be 1g, it could be higher at the equator, but will still be zero at the poles.

 

[Thunderfoot]

In later versions of the Hollow Earth we found out that Mystra was on the opposite side, near the pole there was a "ephemeris area" where you passed through to the other world. It took care of "most" of the physics issues (since it wasn't actually a hollow earth but a Niven Ring like El Mahdi was describing, though the cosmology was viewed from the perspective of a "whole earth" on both sides of the strip. The important thing was that no where did land touch land, it was always water to water so only sailors ever passed through without the aid of magic.

Of course, this was all done much, much, much later in the TSR era in an effort to shut people up about the odd physics of Hollow Earth, and a way to tie the BECMI worlds together (i.e. they were running out of money and were trying to consolidate.)

Frankly, it's fantasy, you can always say...it's magic.

 

[havard]

In later versions of the Hollow Earth we found out that Mystra was on the opposite side, near the pole there was a "ephemeris area" where you passed through to the other world. It took care of "most" of the physics issues (since it wasn't actually a hollow earth but a Niven Ring like El Mahdi was describing, though the cosmology was viewed from the perspective of a "whole earth" on both sides of the strip. The important thing was that no where did land touch land, it was always water to water so only sailors ever passed through without the aid of magic.

In the case of Mystara's Hollow World, it relied on the idea of a gravity field running through the planet's crust. I don't think you are correct about no land touching land. It was all magic, which works fine in a fantasy setting, and most of us never bothered about this.

I did find the above thoughts about visual ranges etc to be interesting though.

Of course, this was all done much, much, much later in the TSR era in an effort to shut people up about the odd physics of Hollow Earth, and a way to tie the BECMI worlds together (i.e. they were running out of money and were trying to consolidate.)

Unless I am not understanding you correctly, this is not correct. The Hollow World Setting was tied to the BECMI Known World
(ie part of Mystara) from the beginning.

Unless you are referring to some mention of Hollow Earth type campaigns in D&D prior to the release of the Hollow World boxed set?

Frankly, it's fantasy, you can always say...it's magic.

Exactly. IMO TSR was often much to preoccupied with coming up with physics explanations for things when it really never was needed. Probably because many of the designers coming from a science background

-Havard

 

[Kaodi]

In the idea for a campaign using hollow worlds, it is actually explicitly magic, though this would not necessarily be known to the players at the beginning.

My idea was that an ancient dwarven civilization had crippling agoraphobia brought on by the realization of the vastness of the sky and space. So they began to research powerful magics that would allow them to create their own worlds that would be more suited to their fears. Their solution was to create an indefinitely large series of hollow worlds in which they could live in. Of course, not everyone in these worlds remembers all of this: in some worlds maybe they ended up living underground again, and in others they live out in the open like other races, and each world is populated differently depending on size and luck (these worlds range from very large to very small). All of them are connected by portals, though some special worlds may have multiple layers that can be traversed through other means.

I was thinking that I could start a campaign with a world that had dwarves (who lived underground), humans, half-elves (just called elves by the people of this world), halflings and warforged (who might be hidden from the other races at the beginning, and were originally constructed by dwarves to be their servants). The first one would be fairly substantial in size, but not bigger than one of our continents at most.