Conservation of Angular Momentum in the BWAWCSAPAWBWSS field

Referring to the verse description of Eliera, it’s been an on-and-off project of mine to try and actually formalise equations for the BWAWCSAPAWBWSS field, but in doing so I have to request the word of writer god on an issue I’m facing - does the orbital plane of a satellite in low Elieran orbit track inertial space?

By which I mean that suppose we accept that a satellite in low ‘circular’ Elieran orbit can maintain a constant altitude above the distinctly coin-shaped surface of Eliera, does its orbital plane similarly rotate with the planet, such that its ground track stays constant? Or are there the equivalent of polar orbits on Eliera, such that the satellite’s trajectory always maintains the same plane with the fixed stars, and follows the shape of Eliera but with wildly varying orbital shapes over the course of an Elieran day, ranging from thin ellipse to full circle?

I’ll note that I haven’t done any math on this as of yet, and it’s highly possible that whatever you choose now may end up being completely infeasible even with Precursor tech or break some key bit of worldbuilding, in which case maybe we can all agree that you get one free bailout retcon on this?

I think the precursors do get a free pass on conservation of energy, maybe somewhat less on conservation of angular momentum; conservation of energy implies the laws of physics being invariant over time, which we know isn’t true with ontotech, so conservation of energy doesn’t have to apply (although, come to think of it, the mystery matter could well include some precursorish energy generation (I seem to recall a self-powered library lamp with no obvious energy source)). Conservation of angular momentum implies the laws of physics not caring about rotation, which hasn’t explicitly been violated on-screen yet.

It’s quite possible that actually doing the math for orbital mechanics requires nontrivial amounts of calculus and numerical integration - there’s also the meta-law smoothing between the precursor field and genuine gravity.

I think our author most certainly gets a free bailout retcon for this.

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My naive stab at this is to start with the equations of motion for a normal planet, and then just apply some spherical harmonic function to squash the entire space until the planet is shaped like Eliera, and observe how many magic forces arise as a result of that.

I do however want to avoid violating conservation of energy as much as is possible in such a scenario. Certainly godlike Precursors could wave their hands and manifest energy on the spot, but I’m a big proponent of Karl Schroeder’s take on Clarke’s Third Law - if the BWAWCSAPAWBWSS field does exist, it probably wants to do as little of its own work as possible.

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This, to a large extent, is something I have tried to avoid being too specific about yet (pending the time and will becoming available to write Magical Flat Planet Orbit Simulator, which sounds way too much like a narrative JRPG or satgirl dating game), because as it turns out, in my head, I’m not actually all that much of a shape rotator, or at least a complex multiple shape rotator.

I, too, have considered these two options. (There is also the issue of orbits around Eliéra also needing to precess in accordance with the second element of Eliéra’s rotation; i.e., that which keeps its primary spin axis at a tangent to its orbit, but blah blah tidal effects blah. That one is tiny compared to the others.)

I don’t like the frame-dragging (i.e., make the plane of polar, or rather parallel-component [1], orbits rotate with the planet) solution for a couple of reasons. Firstly, it requires introducing yet another phenomenon to make it happen, and I prefer not to multiply phenomena beyond the demands of necessity coolness; and secondly, it can’t be made to decay neatly like the other one, since all parallel orbits that would otherwise intersect the planet need the same frame-dragging force applied to them or else fall right out of the sky, and physical phenomena tend not to have “same intensity to this point, then go away”, or even “…and then decay” curves to 'em.

Also, it means I can’t have an equivalent to the track-variance of polar orbits, and I like the track-variance of polar orbits.

But I don’t think non-rotating orbital planes (i.e., not rotating with the planet’s spin modulo the blah blah tidal effects given above) are should be all that problematic. By which I mean: orbits naturally follow isogravs, rather than fixed altitudes, which you can see from the effect of mascons. (Especially on Luna, which has rather dramatic mascons for its mass, giving it a rather irregular gravitational field.)

Since the effect of the [for brevity] This Is Some Bullshit field is to distort Eliéra’s gravitational field shape, a non-maneuvering satellite in parallel orbit should follow the isogravs around the planet, and as those isogravs move as the planet spins, it’ll just keep following them. The effect on the orbital shape (and distance, and period) will be much more dramatic in low orbit than in high orbit, where it ultimately smooths out into the familiar near-circular ellipse, but I don’t think it breaks at any point.

(I don’t think this introduces any conservation of energy problems over and above those you get from the mere existence of a continent-sized machine capable of playing origami with space-time whose power source remains unspecified. Assuming the space-time comes pre-bent, objects in free fall still just fall freely.

And yes, my dear sweet gods, the local orbital mechanics math is painful. There’s a reason they were late to space, after all.)

[1] For convenience, I call equatorial-equivalent orbits “perpendicular” orbits, since they are, like equatorials, perpendicular to the spin axis; by the same nomenclature, pure polar orbits become “parallel” orbits.

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Tend to agree with the points you’ve laid out, except that even without violating CoE you can get lots of funkiness at play that cannot be sufficiently described by a pure potential field. My hunch at this point is that you’re going to get something akin to a gravitomagnetic field, which does no work but does help bend the orbits into compliance.

With your permission I’ll turn this post into a place to catalog my progress in this area!

Absolutely, go ahead. I’ll be very interested to see what you come up with!

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I’ve done a not-very-good sketch of a Magical Flat Planet Orbit Simulator, and I think an orbit that’s a fairly constant height above datum, as seen on screen, doesn’t work with the assumptions I made. I assumed that the Precursor gravity points straight towards the faces, and that the gravity at the edges is all natural. The end result was that any low enough trajectory that didn’t hit the surface while passing across the faces ended up not being curved enough by the natural gravity as it passed around the edges to get into a stable orbit.

I think stable orbits are going to be really sensitive to the conditions around the edges - that’s where you need to go from going one way across the “heads” side to going the opposite way across the “tails” side, and in not very much space at all. Maybe it could work if you really messed around with the gravity there, but I think it’s going to require more centripetal acceleration than you can reasonably come up with. Additionally, orbits at anywhere near a constant height above datum aren’t going to work for the faces - your path is pretty much straight, so there’s no centripetal acceleration to match acceleration due to gravity.

I think the shape of the gravitational field is going to have to be really weird to support orbits with a constant height above datum - it’s going to have to be really weak when crossing the faces and really strong as you round the edges.

Then again, I think I might be missing some requirement for centripetal acceleration somewhere - perhaps if your orbit axis lined up with the axis of rotation, the rotational movement of the world would provide enough centripetal acceleration for a surface-skimming orbit.

But that would make orbits where the axis orbit didn’t line up well with the axis of rotation impossible.

your sketch doesn’t seem viewable on my browser, maybe it’s the country I’m viewing from; in any case making a massive u-turn around the edge is definitely not within the purviews of normal gravity. I’m thinking something more along the lines of “unexplained Precursor force” that behaves much like the Coriolis force would for a rotating observer - it’s velocity-dependent, does no work (which saves on power generation in the Mystery Matter), but deflects objects travelling at notionally orbital velocity non-inertially.

First though, I’m trying to find a way to conformally map a sphere onto an ellipsoid, as that would preserve gravitational direction nicely; any advice in this area is welcome!

I can’t see the post in question either, FWIW.

In random other notes, it’s worth noting that Eliéra isn’t a perfectly flat disk, it’s lenticular enough to make it look flat given the refractive index of its atmosphere with gravity pointing radial to the nominal surface, so you’ve got some play there.

Also, I don’t think core canon, vis-a-vis random-stuff-I-might-have-said-sometime or revisable drafts, excludes the notion that orbits might be somewhat humped as they pass across the “faces”. I can live with tweaking the function somewhat in that area to produce adequately flattened ellipses and declaring that local orbital mechanics doesn’t like constant-height low orbits.

Ah, I think I forgot to hit “publish” on that - maybe try again in a few days.

I’m not sure a conformal map is the right tool to be reaching for - while direction is important, so is the amount of gravity - and I don’t think a conformal map would preserve that.

In terms of its shape, just how not-flat is it? The refractive index of Earthly air is something like 1.003, so there shouldn’t be much actual curvature.

One possible orbit would have the satellite travelling perpendicular to the faces as it passes the edge, and parallel to the faces as it passes the middle of each face, but this would require some fine-tuning of just how fast the Precursor gravity fades away and how it’s mixed, and how it behaves at the edges. There would be the interesting wrinkle that this orbit comes closest to the surface at the edges and is pretty far from the surface (farther than a natural planet’s equivalent of the orbit) as it passes over the faces.

I think one of the necessary conditions for an orbit is that the path integral of the magnitude of the acceleration due to gravity is equal to the path integral of the magnitude of the centripetal acceleration as you move around the path (and probably between any pair of antipodal points in the orbit, too), which I think would make constant-height orbits impossible.

And yes, as far as I can tell, official canon has only committed to Eliéra being flatish, and that its size is some thousands of miles across.

A conformal map allows me to do two things:

  1. I can work backwards, deriving EOM from a squished version of an elliptical orbit, which sounds way easier than trying to brainstorm some EOM that creates proper orbits, especially when I don’t know (and don’t believe) that conservative gravity alone can square this circle.

  2. I can preserve the direction of surface gravity to be perpendicular to Eliera’s surface at all points, which is canonically established.

As for the actual dimensions of Eliera, it’s stated in the verse description to be a 10000 x 200 mile ellipsoid, and that gravitation defaults to nearly Newtonian at 20 major radii from the planetary center. @avatar Unfortunately it also does state that low orbits neatly maintain altitude above datum, which I don’t think is a deal-breaker by any means yet.

I didn’t think I was that exact - that’s this bit, right?

The practical result of this is that if you are in low Eliéra orbit, say a 10,100 mile orbit (i.e., 100 miles above datum), your stable orbit will skim the atmosphere in what is basically a disk shape orbit matching the gross shape of the “planet”.

…if so, I hereby declare that “basically a disk shape” in this case means “super-duper-squished ellipse”, on the grounds that I think that was what I meant, and if that wasn’t what I meant, it probably should have been - on the grounds that orbits should be orbitiform. Left too much room for interpretation in that “basically”.

Update: conformal maps will NOT solve this problem, seeing as the few that work in 3D geometry tend to explode with increasing distance from Eliera, which is the opposite of what we want. Looks like I’m going to have to do the “add duct-tape fields until it works” approach.