Theoretical Moon Question.

[Drakheim]

For simplistic sake, I will use the Earth as the modle. Could a planet such as Earth have a moon that was in a Polar Orbit. A.K.A. It rises in the North and sets in the South, or vice versa?

[astromark]

Yes. I do not understand you asking. We have polar aligned satellites. It does not seem to happen naturally but, it could happen with a moon.
Orbital capture,. Remembering it is the orbital mass that gravity attracts, or orbits. Not the planetary orientation.

[Drakheim]

Well, to clarify on that then. All the natural moons we have in this solar system (that I am aware of) orbit their parent bodies’ equators. So for example, Uranus’s moons from our view-point orbit what appears to be nearly North to South, even though from the view point of someone on a theoretical surface of that planet would see them going East to West.

That I understand is a general model for natural body orbits. I was curious about a natural body forming a stable polar orbit. Of all the places I got this from was an online game. Everquest II.

The planet it takes place on has two moons. One is a regular moon, although shattered and an unpleasant place to be at the moment. The other moon you can see in the sky is shown on a model in the game a having a North to South orbit of the planet.

[tony873004]

This is not stable. If you moved Earth's Moon into a polar orbit with the same distance it currently has, it will crash into the Earth within a decade.

[trinitree88]

This is not stable. If you moved Earth's Moon into a polar orbit with the same distance it currently has, it will crash into the Earth within a decade.


tony. I would doubt that. That's an awful lot of angular momentum and kinetic energy to shed in ten years. It might create some strange tides, but without any gravity simulator I'd venture a billion years, or more.:shifty: Pete.

[grant hutchison]

There are a lot of outer moons in orbits that are steeply inclined to their parent planet's equator.
Most notably for the question here, the nine known outer Uranian satellites have orbits that are tilted at various angles between 10 and 50 degrees to the ecliptic plane.
They're far enough out that their orbital planes precess more or less in the ecliptic. Meanwhile, Uranus' rotation axis points just 8 degrees south of the ecliptic. So all of these satellites, as their orbital planes precess, will at some time pass directly over Uranus' north and south poles, as they flip from a retrograde to a direct orbit, or vice versa, relative to Uranus' equator.
But according to JPL (http://ssd.jpl.nasa.gov/?sat_elem) the relevant precession periods are measured in millennia, so we're unlikely to see that happen any time soon, unfortunately.

Grant Hutchison

Edit: Looks like Sycorax is currently the closest to being polar, with an inclination to Uranus' equator of 82 degrees in 2000.

[tony873004]

tony. I would doubt that. That's an awful lot of angular momentum and kinetic energy to shed in ten years. It might create some strange tides, but without any gravity simulator I'd venture a billion years, or more.:shifty: Pete.

It doesn't need to shed its angular momentum. It just needs to be perturbed into an orbit with the same semi-major axis, but an eccentricity so large that its perigee is below Earth's surface. The Kozai mechanism will cause this to happen.

Here's a link to a simulation I performed of this exact scenerio: http://www.orbitsimulator.com/gravity/articles/kozai.html
It only takes the Moon 8 years to slam into Earth.

Also, here are links to 2 Kozai mechanism calculators I made.

http://orbitsimulator.com/formulas/emax.html
http://orbitsimulator.com/formulas/pkozai.html

The first calculator computes the maximum eccentricity an object will gain from the Kozai mechanism. Simply place in 90 degrees for inclination, and it will compute that the maximum eccentricity, which in this case will be 1.

The next calculator computes the period of the Kozai cycle. Although designed for binary stars, you can consider the Earth as a star, the Moon as a planet, and the Sun as the companion star. Enter 384000 km for a1 (semi-major axis of Moon's orbit), 1 Earth mass for m0,the mass of the primary star, 0.0123 Earth masses for m1 the mass of the planet, 1 solar mass for m2, the mass of the secondary star, 1 AU for a2, the semi-major axis of the secondary star, and 0 for the eccentricity. You should get an answer of about 13.4 years. This is the period of an entire cycle, from 0 eccentricity to max eccentricity (in this case, 1) and back to 0 eccentricity. So half of this, roughly 7 years, is the time it would take the Moon in a polar orbit to achieve an eccentricity of 1. This is in rough agreement with the simulation, where the Moon took 8 years to slam into Earth.

The formulas provide only estimates (note the ~ sign instead of an equal sign). And in one of the two papers whose links appear on the calculator page (**edit, one is a dead link now.. sorry), it says that The Kozai mechanism can be negated if other forces dominate. I believe this is why the moons of Uranus are not affected. Uranus' equatorial buldge provides a torque that overwhelms the Kozai mechanism, which at Uranus' distance is much weaker than at Earth's distance.

[Ronald Brak]

Cool. Would it break up as it made close approaches to the earth and start forming a weird ring prior to pounding into the planet?

[tony873004]

Cool. Would it break up as it made close approaches to the earth and start forming a weird ring prior to pounding into the planet?

Probably not. The eccentricity is accelerating pretty rapidly towards the end, and it only crosses Earth's Roche limit (~4 Earth radii, I think?) a couple orbits before impact. Being at perigee, it rushes through the Roche zone, so any matter that levitatited off the Moon's surface would be pulled back to the Moon as it quickly exits the Roche zone.

[grant hutchison]

... it says that The Kozai mechanism can be negated if other forces dominate. I believe this is why the moons of Uranus are not affected. Uranus' equatorial buldge provides a torque that overwhelms the Kozai mechanism, which at Uranus' distance is much weaker than at Earth's distance.It's likely that I'm missing something, here.
But it seems to me that the torque from Uranus' equatorial bulge must be small in comparison to solar torque for these satellites, otherwise the Laplacian plane wouldn't be so close to the ecliptic.
Is it possible that they just have a Kozai cycle that never gets eccentric enough for them to hit Uranus? (I've no intuitive grasp of how likely or unlikely that is.)

Grant Hutchison

Edit: Sorry, it belatedly occurs to me I could do the sums myself on your calculator or from the paper you link to.
I'm pressed for time and distant from my usual references right now, but I'll get to it ...

[Ronald Brak]

The movie script practically writes itself. The earth captures a new moon the size of New Mexico in a polar orbit and Bruce Willis has only eight years to detonate the nuclear bomb that will cause it to crash into the lunar surface instead of the earth.

[Maksutov]

Well, to clarify on that then. All the natural moons we have in this solar system (that I am aware of) orbit their parent bodies’ equators....Earth's Moon doesn't. Its orbit is close to the plane defined by the Earth's orbit around the Sun.

[tony873004]

I get a maximum eccentricity of 0.99, and a kozai period of 1 million years for Umbriel. So it should have slammed into Uranus thousands of times by now. But its equatorial satellites are in happy round orbits.

There is a large gap between its equatorial satellites and its irregular satellites. Possibly, the effects from Uranus' equatorial buldge fall off rapidly beyond the orbit Oberon, allowing the Kozai mechanism to dominate. Equatorial satellites beyond this region may have slammed into Uranus a long time ago.

After this large gap comes the irregular satellites, with inclinations as high as 50 or 60 degrees to Uranus' orbital plane. The Kozai mechanism may be responsible for the lack of satellites with inclinations higher than 50 or 60 degrees. The Kozai mechanism would cause their periapses to drop into the region dominated by Uranus' large moons. At least that's one of the theories for why Jupiter has no moons with inclinations greater than 60 degrees.

Here's a diagram of the Uranus system. It's viewed from the ecliptic plane, so you're looking towards Uranus' north pole (or south pole, I get confused). The top diagram shows its inner satellites in their round orbits. The bottom diagram shows the full Uranian system.
http://orbitsimulator.com/BA/uranus1.GIF

[grant hutchison]

Interesting. Thanks, Tony.

Grant Hutchison

[EDG]

Wow, that's a bit of a mess! :)