Astronomy question

[Nyeshet]

Aristotle did not have a defined distance for each of the crystal sphere; he just had the setup. Ptolomy came up with distances, but they tended to vary widely due to his 'circles within circles' type orbits - used to explain retrograde motion and some other anamolies. I suggest the following link:

http://www.phys.uu.nl/~vgent/astro/almagestephemeris.htm

Playing with it for a while, I found that:

- Beyond Saturn: the Sphere of Fixed Stars (starts ~20,000 earth radii away).
- Saturn ranged from about 14,700 to 19,600 earth radii distant, ~17,150.
- Jupiter ranged from 8,800 to 14,200 earth radii, ~11,500.
- Mars ranged from 2,100 to 8,500 earth radii, ~5,300.
- Sun ranged from 1,160 to 1,260 earth radii, ~1210.
- Venus ranged from 160 to 1,170 earth radii, ~665.
- Mercury ranged from 69 to 173 earth radii, ~121.
- Moon ranged from about 54 to 66 earth radii distant, ~60.
- The Earth, ~1.

Do you notice how one 'Sphere' ends where another begins? That was devised on purpose. From what I read, Ptolomy noticed something like this between two of the 'spheres' (a parallax of merc-venus, I think) and decided such symetry was too perfect to be accidental, so he made the necessary adjustments here and there in his system until he had something that both gave more or less correct mechanics for the motions in the sky while also holding this 'symetry' of distances.

It's not quite perfect, however. Note the gap between the Sun's sphere and Mars' sphere. Note the slight overlap in two instances (although I'll admit I was using the +10yr button a few times, and it was the perturbation of minor errors over centuries that eventually brought sufficient doubt to the Ptolomaic system as to allow the heliocentric theory a chance (at least prior to Kepler and Newton, who affectively nailed the lid shut on its coffin).

Note also that Ptolomy did not have impenetrable crystal spheres. Instead he had lots of circles within circles that effectively made all the spheres semi-permiable and Nested within each other. (Saturn, for instance, had an equant in the Sphere of Mars! ) Also, each 'sphere' had two layers: a layer of fire ("above") and a layer of air ("below").

Also note that the earth itself was supposed to be four layers: Earth, Water, Air, and Fire (in ascending order). So the oceans definately had Earth below their watery surface - eventually. The continents, by corollary, must be viewed as islands sticking out of the sphere of Water-Earth. This 'sphere' of "Water-Earth" was then nested within the 'sphere' of Fire-Air. Perhaps he used heat haze to explain twinkling?

If you wish to make the other worlds inhabitable, they would have to be in the lower half of the equivalent sphere (in the air layer between the fire layer of the prior nested sphere and their own fire layer).

 

[The Sigil]

Circular orbits are fine. Even preferred, actually.

Okay, this actually makes things pretty easy.

The angular size of an object ('a') as a function of distance to the object ('d') and the radius of the object ('r') is expressed by:

a = 2 tan^-1 (r/d)

However, since you have said that you want the angular sizes (i.e., how they appear to the earthbound observer) to be the same, we can get rid of all the nasty inverse tangent functions and come up with:

r1/d1 = r0/d0 or, rearranged slightly, d1 = d0*r1/r0

Where r1 is the fictitious planet's radius, r0 is the real planet's radius, d0 is the real distance to the planet, and d1 is the distance to the fictitious planet.

To make things as simple as possible, I'll assume all "planets" are exactly earth-sized; doubling the planet's size doubles the distance to the planet, and is a trivial exercise that can be done at home.

From here, it's a simple matter of taking average distance from earth to a planet (which can vary by quite a bit, but since you just wanted first-order approximations, we'll do that) and actual radius of a planet (expressed in "earth radii"). Since we're using "earth radii" as units of measurement, and we have said all fictional planets are exactly earth-sized, r1 = 1. Our formula for distance is then simply:

d1 = d0/r0

Here we go:

Sun - 1.37 million km (radius is 109 earths, average distance is 150 million km)
Moon - 1.41 million km (radius is .273 earths, average distance is 384 thousand km)
Jupiter - 69 million km (radius is 11.2 earths, average distance is 778 million km)
Saturn - 150 million km (radius is 9.45 earths, average distance is 1426 million km)
Venus - 158 million km (radius is .949 earths, average distance is 150 million km)
Mercury - 391 million km (radius is .383 earths, average distance is 150 million km)
Mars - 422 million km (radius is .533 earths, average distance is 225 million km)
Uranus - 715 million km (radius is 4.01 earths, average distance is 2870 million km)
Neptune - 1160 million km (radius is 3.88 earths, average distance is 4500 million km)

NOTES:

On a strict average over infinite time, every planet "outside" us is the same distance from us as it is from the Sun and every planet "inside" us is the same distance from us as the Sun is. I used semi-major orbital axes for all "average distance" measurements except earth-sun and earth-moon as these were more easily found on Wikipedia.

This number of roundings and huge oversimplifications (going from elliptical orbits to spherical orbits, assuming constant distance between the planets instead of varying distance due to different orbital speeds around the sun) means this system is GREATLY simplified... but is a "first order" approximation for you.

Luminosity equations (not shown) tell us that the Sun needs to be at the same temperature as it is now when reduced in size and distance to give the same amount of energy to the earth. (Chopping the size reduces the energy given off by the "reduction factor" squared, but bringing it closer reduces the energy needed by the "closer factor" squared; these wind up cancelling out)... so the Sun doesn't have to be any "hotter" than usual on the surface to provide the same energy to the earth.

Again, if you want to move planets farther out/closer in, you just enlarge/reduce them by a factor of the same amount; for instance, if you want the Sun to be twice as large as earth, move it twice as far away (to 2.74 million km out) to keep things the same. If you want the moon to be half of earth size, you can move it in to half its listed orbit (i.e., to about 700 thousand km out).

 

[Baron Opal]

Sigil, Nyeshet, thank you both very much. It's appreciated.

 

[Psion]

sephiroth is plural already

Yeah, he had like, what, 7 clones.

 

[fusangite]

Note also that Ptolomy did not have impenetrable crystal spheres. Instead he had lots of circles within circles that effectively made all the spheres semi-permiable and Nested within each other. (Saturn, for instance, had an equant in the Sphere of Mars! ) Also, each 'sphere' had two layers: a layer of fire ("above") and a layer of air ("below").

But it's important that we don't apply modern principles of scientific thought to this model. Aristotle's work was in the discipline of Physical Astronomy (how the universe is) whereas Ptolemy's work was in the discipline of Mathematical Astronomy (how the universe works). Ptolemy's calculations were to say "the universe works as though it's made like this"; Aristotle's a priori assertions were to say, "this is the true nature of the universe."

Thus, while everybody had to use epicycles, retrogrades, etc. to do calculations in Ptolemy's model, everybody doing these calculations also believed the spheres were impenetrable. Basically, the mathematical model was a lower grade of truth (true nevertheless for its own limited purposes) than the philosophical model.

Thus a Ptolmaic astronomer in the Middle Ages and, probably even Ptolemy himself would have happily credited that the spheres were impenetrable even while using epicycles and the like to make predictive calculations.

Also note that the earth itself was supposed to be four layers: Earth, Water, Air, and Fire (in ascending order). So the oceans definately had Earth below their watery surface - eventually. The continents, by corollary, must be viewed as islands sticking out of the sphere of Water-Earth. This 'sphere' of "Water-Earth" was then nested within the 'sphere' of Fire-Air.

But these were never conceptualized as spheres at all. The point is that everything in the lowest sphere was made of corruptible elements whereas the spheres beyond were made of the fifth, perfect element, quintessence. The perfection of the quintessence is the reason the spheres were physically impenetrable by the four lowly terrestrial elements and why the only entities that could exist beyond the spheres were angels.

If you wish to make the other worlds inhabitable, they would have to be in the lower half of the equivalent sphere (in the air layer between the fire layer of the prior nested sphere and their own fire layer).

No. Absolutely not. Nothing made out of terrestrial elements exists outside the sublunar sphere.

And, by atomic, I'm refering to the scheme where atoms have a descrete center (Earth / Nucleus) and then the orbiting masses (Planets / Electrons) traveling in distinct regions (Crystal Shells / Electron Shells). At the moment, I'm thinking of having each planet being a mini-transitive plane or the top layer of a plane. The number of layers is equal to the ideal (noble) number of electrons for that atomic shell.

This is not Aristotelian per se in that:
(a) there is only one earth and set of spheres in his theory of the universe
(b) the properties of fixity and incorruptibility are much more appropriately applied to the nucleus in the atomic model whereas in the Aristotelian model, these properties are best applied to the outer spheres
(c) if the area outside the sublunar sphere is inhabited, it is logically necessary that it be inhabited by perfect beings and not be earth-like ie. corruptible

That's not to say I don't like your model but it seems fair to say your proposed universe is Ptolemaic rather than Aristotelian.

Also, I've seen drawings of multiple Trees of Life where each iteration is farther and farther away from God.

I think in the Norse and Aztec ones, the gods are on top and we're in the middle but I'm betting that because of Platonic influence, the Cabalistic models are more likely to put us on the bottom and God on top. If you're interested in vertical cosmological schemes like this, I highly recommend the 1950s book (now outdated) book on Nahua (Aztec) mythology called Firefly in the Night. Nicholson's inappropriate rationalization of their cosmological scheme sounds like a very nice fit with what you're doing.

The qlippoth get involved here, and I'm not too sure what it / they are. So, by travelling up through the crystal shells and finding the Crown sephiroth, which is the exit point (say, Polaris) to the next iteration of the Tree. That gets you to the next atomic nucleus / Earth / what have you. So, you could eventually sail your way to the Prime Mover if you have the wherewithal.

Here, you're doing what many medievals and early moderns did: hybridizing Aristotle and Plato. Just be aware you're doing it. The Cabalistic/Gnostic/Sufic idea of emmanations assumes that as things get closer to God, they get less corporeal and that the physical world is so intrinsically inferior that no physical travel could get you to such a destination.

Anyway, have fun!

 

[RangerWickett]

I can't provide mathematical help, but I will say this discussion is fascinating. In my own campaign I just had a traditional, modern concept of a solar system. It just so happened that every planet was inhabited with life (though not necessarily intelligent life), and was in truth one of the 'planes.'

I took a vague model of our solar system.

Mercury - Plane of Fire
Venus - Plane of Air
Earth/Campaign Setting - Plane of Life (and the moon was the Plane of Dreams)
Mars - Plane of Water
Asteroids - Plane of Earth
Jupiter/Saturn - Plane of Space
Ring around Jupiter/Saturn - Plane of Time
Pluto - Plane of Death (and its moon was the Plane of Ruins)

The only way to get to another 'solar system' of new planes would be to planeshift through the sun, because suns were like fiery wormholes between different planar systems.

I set up that cosmology because I thought it'd make for interesting planar adventures if they ever came up, but I never really considered what scholars and layfolk thought about it, and how their believes and perceptions might have been wrong or biased.

Again, a very interesting thread.

 

[Wombat]

Find yourself a copy of Richard Garfinkle's Celestial Matters ASAP!

http://www.amazon.com/exec/obidos/t...103-7705929-2234214?v=glance&s=books&n=507846

This is a "hard sci fi" book where the physics being used is Aristotelean ... and the protagonists are sailing to the Sphere of the Sun to cut a chunk off as a weapon! Great background material for such a game, and a whopping fun read to boot!