Brain Break: Anyone Willing to entertain another (strange?) gravity / orbital question?

[JoeSchmuckatelli]

The question:  what would an orbit look like for a satellite around a non-spherical planetary or stellar mass object?

i.e. if instead of a spherical planetary mass object - we had a cylindrical mass, the length of which was equal to the diameter of a planet or star?

• would every orbital path around such an object be elliptical?
• If a circular orbit were possible, how?

 

Caveats:

I know why objects over a certain mass tend to be spherical (c.f. hydrostatic equilibrium)

I also know that with a spherical body we can / do think of it as a point mass, because regardless of where the satellite is in relation to the planet or star, a line drawn from the satellite through the center of mass will measure the same diameter of the sphere in any direction

I suspect that we cannot think of a cylindrical object as a point mass, because if you drew a line from the orbiting object through the center of mass of the cylindrical object, you get different amounts of 'stuff' depending on where the satellite is in relation to the cylinder...

 

----- I thank anyone willing to chit-chat with me about this stuff; stuck at home with the kids and this kind of thinking is my 'brain-break' from 6th grade social studies discussions with my 12 y.o.!

• also, missing the 'good ol' days' when I could meet up with college friends and babble idly about science over a beer.  

[cubinator]

Well, you could have an infinite cylindrical mass and have a moon orbit around it in a spiral. I think a finite cylinder would just cause the moon to be ejected once it reached the end. Here's something really fun though: toroidal planets.

A donut planet would theoretically be gravitationally stable, if you could get it to form in the first place (basically, you spin it up so fast that the centripetal force balances out gravity and you essentially form the planet in its own GEO) and it gives rise to some interesting properties, which I coincidentally explored with friends in college late at night on the back of an actual napkin. Here's what we came up with. But first, a visualization:

This planet has a small moon which orbits in a figure-eight around the hole of the donut, seen in the third illustration. You could also have a moon in an "orbit" that just goes up and down through the middle of the hole. Both of these options are probably quite unstable.

Some other interesting things about the donut planet that aren't explicitly related to orbits:

Mapping is nice. Not only does the left side cross onto the right on a rectangular map, but the top and bottom are connected too! There's still some distortion around the edges, but you can get around that by having separate maps of the outer and inner regions.

The outside and the inside of the donut, which we named the Outer and Inner Circles, respectively, have some distinguishing properties. They are separated by the North and South Circles, which are analogous to Earth's poles except that they are long loops instead of spots. Seasons and daylight on the Outer Circle work just like they do on a sphere (for reference, we decided to give this planet about 45* axial tilt, you'll see why it needs more than Earth in a moment). 

Crossing over to the Inner Circle, though, we encounter some very odd changes. First of all, timezones are phase-shifted by half a day! So you'll have to change your clocks by 12 hours going over the poles. Nighttime occurs when the Sun is behind the ground, but a lot of the ground is actually in the sky now! That's problematic because if the planet has low axial tilt, most of the Inner Circle won't ever get any sunlight! This is why we gave the planet high axial tilt. The Inner Circle is still in darkness during the spring and fall equinoxes, and thus is likely much colder than the Outer Circle.

Of course, when you're in the Inner Circle you can see the other side of the donut hole above you. The horizon actually has negative curvature when you're looking to the equator, and positive curvature when looking to the poles! During nighttime you can see the part of the Inner Circle that's in daylight above you, so most of the time the night is quite bright. 

This effect is even noticeable in the Outer Circle. The polar radius of the planet is quite small, even smaller than Earth, but the equatorial radius is enormous. Thus, if you're standing on a mountain looking around on a clear day, you'll be able to see extremely far in the equatorial directions, but a very short distance in the polar ones. Standing on the inner edge of the poles, you'd be able to see that walking longitudinally you'd go in a big circle.

[Bill Phil]

Well the truth is we can’t treat real planets as spherical - they’re not.

Terms like the J2 term are used to describe unequal mass and geometrical distributions as alterations to the point mass model. The J2 term specifically models a planet’s oblateness - which gives rise to precession of the ascending node. The ISS’s orbit precesses around 5 degrees a day (roughly 16 orbits) as a result. Precession around Jupiter or Saturn would be even more extreme.

I suspect you could model the cylinder as a line of discrete points - perhaps the gravitational field could be calculated analytically using calculus though I can’t say for sure.

Then integrating with a runge-kutta or symplectic integrator would let you see how that changes trajectories.

Actually, I might do that myself...

[Bill Phil]

1 hour ago, cubinator said:

Well, you could have an infinite cylindrical mass and have a moon orbit around it in a spiral. I think a finite cylinder would just cause the moon to be ejected once it reached the end. Here's something really fun though: toroidal planets.

Well, not necessarily. Around a finite cylinder the moon would still experience a gravitational force toward the cylinder even past the ends.

In the case of our Moon, it's so far away from the Earth and the Earth is very small compared to that distance - so the real difference is likely insignificant to the Moon's orbit, or at most only perturbs it slightly.

[cubinator]

I think it's possible to lock planets in place in Universe Sandbox. You could simulate an approximate cylindrical planet by placing many planets in a line and throwing moons about.

[JoeSchmuckatelli]

https://www.goodreads.com/book/show/939740.The_Integral_Trees

 

(still reading the replies but thought it might throw this out there) 

 

Interesting to see people thinking about the satellite orbiting the circumference of the cylinder... I was trying to visualize an orbit around the ends (going from end to end) of the cylinder - which I know would not happen naturally or be stable... 

 

More of a thought experiment than anything 

10 minutes ago, cubinator said:

... 

Have you seen Niven's story linked above? 

Edited March 26, 2020 by JoeSchmuckatelli

[magnemoe]

23 minutes ago, Bill Phil said:

Well the truth is we can’t treat real planets as spherical - they’re not.

Terms like the J2 term are used to describe unequal mass and geometrical distributions as alterations to the point mass model. The J2 term specifically models a planet’s oblateness - which gives rise to precession of the ascending node. The ISS’s orbit precesses around 5 degrees a day (roughly 16 orbits) as a result. Precession around Jupiter or Saturn would be even more extreme.

I suspect you could model the cylinder as a line of discrete points - perhaps the gravitational field could be calculated analytically using calculus though I can’t say for sure.

Then integrating with a runge-kutta or symplectic integrator would let you see how that changes trajectories.

Actually, I might do that myself...

The moon has plenty of places with higher gravity because of metal asteroid impacts so an low orbit tend to be unstable. Earth has this too but as its much larger and has an heavy iron core the gravity variations is far smaller than atmospheric drag. 

An doughnut planet is stable but idiotic unlikely to form naturally. 
However any orbit passing trough the center would be unstable like the L1, L2 and 3 points. You could put an satellite there but not anything natural. 

[Bill Phil]

Alright I put together a model in MATLAB. 

(I already had a similar one lying around, didn't take much to modify it)

(Hopefully this works)

Imgur album with plots:

https://imgur.com/a/xMHwucs

Assuming a cylindrical planet can be modeled as a line of discrete particles.
Used an RK4 integrator with a step size of 60 seconds.
Number of particles is 100. Each with about 1/100 the mass of Earth.

Looks like it acts like mass concentrations around the poles.

Edited March 26, 2020 by Bill Phil

[SuperFastJellyfish]

3 hours ago, JoeSchmuckatelli said:

if instead of a spherical planetary mass object - we had a cylindrical mass, the length of which was equal to the diameter of a planet or star?

Maybe something like this?

 

Edited March 26, 2020 by SuperFastJellyfish

[magnemoe]

1 hour ago, JoeSchmuckatelli said:

https://www.goodreads.com/book/show/939740.The_Integral_Trees

 

(still reading the replies but thought it might throw this out there) 

 

Interesting to see people thinking about the satellite orbiting the circumference of the cylinder... I was trying to visualize an orbit around the ends (going from end to end) of the cylinder - which I know would not happen naturally or be stable... 

 

More of a thought experiment than anything 

Have you seen Niven's story linked above? 

Integral trees is different, you take something like an super Venus  and put in low orbit around an neutron star. However you would also need nitrogen I assume. (book said an ice giant they whey are mostly hydrogen who would not work)
Now the point here is that the atmosphere will leak out of the planet but would be trapped inside the steep gravity wheel of the neutron star creating an breathable atmosphere in orbit. 
Like an doughnut planet its very unlikely to get created bur something some might want to create for fun as it would be an fun place. 

The integral trees was an sort of tree who was hundreds of km long and stabilized by tidal forces.