Energy of the tilted orbit

[astrochumak]

Hello everyone,
I've been recently looking up some info on Saturn's rings and particularly their thinness, and the following sentence got me thinking:

"Why are Saturn's rings so thin? It has to do with the ring particles colliding with each other. Ring particles that are high above or below the rings are in a highly "inclined" (tilted) orbit, and have more energy than ring particles that are closer to the ring plane. When those particles collide with other particles, some of their energy is lost, so causing them to move to lower-energy orbits closer to the ring plane."

How come the tilt of the orbit gives an object more energy in such way that over time, due to different perturbations, it comes down to some other orbit? 

Link to the original site: https://caps.gsfc.nasa.gov/simpson/kingswood/rings/

[vlaiv]

I have no idea, and generally know a little about the topic, but here is quote from the Wikipedia that might offer some clue:

Quote

In 1966, Peter Goldreich published a classic paper on the evolution of the moon's orbit and on the orbits of other moons in the solar system.^[6] He showed that, for each planet, there is a distance such that moons closer to the planet than that distance maintain an almost constant orbital inclination with respect to the planet's equator (with an orbital precession mostly due to the tidal influence of the planet), whereas moons farther away maintain an almost constant orbital inclination with respect to the ecliptic (with precession due mostly to the tidal influence of the sun). The moons in the first category, with the exception of Neptune's moon Triton, orbit near the equatorial plane. He concluded that these moons formed from equatorial accretion disks. But he found that our moon, although it was once inside the critical distance from the earth, never had an equatorial orbit as would be expected from various scenarios for its origin. This is called the lunar inclination problem, to which various solutions have since been proposed.

To me, this gives a clue that there will be difference if ring material is mostly formed from material in solar system versus material of planet itself and that tidal forces of the Sun and the planet have impact on orbit of material.

It could be that it has something to do with shape of Saturn. It is not perfect sphere because it rotates. It could be that gravitational pull is not the same in orbit that is perpendicular to axis of rotation of the Saturn and those inclined to it.

[andrew s]

You could try this Saturns rings. It's  complicated! The link is to a pdf that downloads. Section 2.2.2 is the relevant one. 

Regards Andrew 

Edited December 29, 2020 by andrew s

[Waddensky]

Hmm interesting. At first glance, I'd say that the gravitational forces of the Sun and the other planets should pull the ring particles into the plane of the orbit, but since the inclination of Saturn's axis (and therefore its rings) is about 27 degrees that couldn't be the reason. Maybe it has something to do with the centrifugal force due to the rotation of the rings?

[saac]

For a simpler approach you could consider how tangential velocity varies with latitude.  Every planet will have an associated minimal orbital velocity (about 7.8 km/s for Earth at equator).  Satellites tend to be launched from the equator in the opposite direction of Earth's rotation as this will give a relative assistance in their tangential speed (approx 0.4 km/s) .   The tangential velocity at Earth surface reduces with latitude in accordance with v = ωrcosϴ   where ω is Earth's rotational velocity (angular velocity), r is Earth's radius and ϴ is the angle of latitude.  You can see then that launch sites at a higher latitude benefit from a lower initial tangential velocity.  Accordingly, a greater amount of energy is needed to achieve an orbital launch for higher latitudes.   So  if we consider that tilted rings around a planet are effectively orbiting at higher latitudes than those of a co planar ring they too will need a higher relative tangential velocity in order to maintain their orbit.  Such rings therefore have a greater energy level than their co planar counterparts.    That's my initial take on it, happy to be corrected

 

Jim 

[vlaiv]

10 minutes ago, saac said:

 So  if we consider that tilted rings around a planet are effectively orbiting at higher latitudes than those of a co planar ring they too will need a higher relative tangential velocity in order to maintain their orbit.  Such rings therefore have a greater energy level than their co planar counterparts.    That's my initial take on it, happy to be corrected

This would be true if orbits were like this:

But inclined orbits are like this:

It still orbits along the equator - just not equator that is along axis of rotation of the planet.

[andrew s]

If Saturn was spherical then all orbits about its centre would be equivalent.  However it has an editorial bulge which tends to pull particles into its plane. A combination of inelastic collisions and the conservation of angular momentum does the rest in a hand waving kind of way.

Regards Andrew 

Edited December 29, 2020 by andrew s

[Tiny Clanger]

10 minutes ago, andrew s said:

If Saturn was spherical then all orbits about its centre would be equivalent.  However it has an editorial bulge which tends to pull particles into its plane. A combination of inelastic collisions and the conservation of angular momentum does the rest in a hand waving kind of way.

Regards Andrew 

Ah, Handwavium, my favourite element on the periodic table 😀

Admittedly it's on the back of the periodic table , but it's an element with so many uses ...

[saac]

4 hours ago, vlaiv said:

This would be true if orbits were like this:

But inclined orbits are like this:

It still orbits along the equator - just not equator that is along axis of rotation of the planet.

It was an inclined orbit I was thinking about vlaiv.   Here , the diagram may show better what I'm thinking - again I may be way off on this.  At the planet's equator (90 degrees from axis of rotation at lat =  0 ) we find the tangential velocity at its maximum value.  Deviate from the equator then as latitude increases the tangential velocity component decreases.  Now along that so called non equatorial orbit the tangential velocity will vary as the latitude changes but I think Im right in thinking that it will always be lower than a corresponding point on the equatorial orbit. So a particle of ice in a tilted orbit will not benefit from as high relative tangential velocity (due to the planet's motion) compared to its equatorial equivalent.  It will need a higher total velocity then to make up the delta to maintain that orbit. From my diagram there are 4 points along the non equatorial orbit which share the same tangential velocity as the equatorial orbit - they possible have a name in geometry - not sure what !    Again this is my take on orbital mechanics -  I certainly wouldn't have won a job on the Hidden Figures movie   

Jim

 

ps just noticed my diagram is wrong - it shows 4 points where my tilted orbit crosses the equator - there should only be 2 !  For clarity Orbital Point 2 is one of many orbit points (ice particles positions) on the titled orbit path.

 

 

Edited December 29, 2020 by saac

[andrew s]

12 minutes ago, saac said:

It was an inclined orbit I was thinking about vlaiv.   Here , the diagram may show better what I'm thinking - again I may be way off on this.  At the planet's equator (90 degrees from axis of rotation at lat =  0 ) we find the tangential velocity at its maximum value.  Deviate from the equator then as latitude increases the tangential velocity component decreases.  Now along that so called non equatorial orbit the tangential velocity will vary as the latitude changes but I think Im right in thinking that it will always be lower than a corresponding point on the equatorial orbit. So a particle of ice in a tilted orbit will not benefit from as high relative tangential velocity (due to the planet's motion) compared to its equatorial equivalent.  It will need a higher total velocity then to make up the delta to maintain that orbit. From my diagram there are 4 points along the non equatorial orbit which share the same tangential velocity as the equatorial orbit - they possible have a name in geometry - not sure what !    Again this is my take on orbital mechanics -  I certainly wouldn't have won a job on the Hidden Figures movie   

Jim

 

 

Hi @saac struggling to understand what your saying . If the blue orbit is just a tilted white EQ orbit how can it intersect at the green points? Can you add an arrow to show what you mean by the tangential velocity? Is it the tangent to the orbits or some other projection.

Regards  Andrew 

[saac]

22 minutes ago, andrew s said:

Hi @saac struggling to understand what your saying . If the blue orbit is just a tilted white EQ orbit how can it intersect at the green points? Can you add an arrow to show what you mean by the tangential velocity? Is it the tangent to the orbits or some other projection.

Regards  Andrew 

Andrew Im thinking that the tangential velocity is referenced against the axis of rotation . The tilted orbit effectively crosses the equator at two instances in its orbit - these were supposed to be shown by the green dots  

Jim

[andrew s]

Ok Jim, it was the 4 green dots that confused me. The tangential velocity is as you describe for points on a rotating solid body.

For an identical inclined free orbit about a spherical planet it will have its axis of rotation perpendicular to the plane of its orbit and have the same angular velocity about that axis.

The maximal launch boost at the equator is due to the rocket being on the earth.

Hope I have understood now.

Regards Andrew

Edited December 29, 2020 by andrew s

[vlaiv]

3 hours ago, saac said:

It was an inclined orbit I was thinking about vlaiv. 

I'm failing to see connection - unless you are implying that ring material was originally part of the planet itself and got ejected from different places?

[Macavity]

Thinking about particles in orbits I was reminded of (LHC) beams in accelerators etc.
"Stochastic cooling"! But a search on "Saturn Rings + Stochastic" DOES yield results. 😉
(N.B. analogies don't always work... Except perhaps in Alchemy? lol)
 

Edited December 30, 2020 by Macavity

[Gfamily]

Another consideration is the mass of the rings themselves, as any particles that are out of the plane will be subject to a gravitational force towards the plane of the rings, which will maximise their speed as they pass through the rings.  

However, I think the more significant factor is that although individual particles each have their own peculiar motion at the outset; as a system, the rings have a net angular momentum - which defines a plane and an axis.

The system's total angular momentum is conserved over time, and at the same time, the inelastic collisions between particles will eventually remove the peculiar components of the motion out of the plane. 

[basingset]

Perhaps it has something to do with the shape of Saturn. It is not a perfect sphere because it rotates. It could be that the gravitational pull on orbits perpendicular to Saturn's axis of rotation and inclined to it are not the same. The gravitational forces of the Sun and the other planets should pull the ring particles into the plane of the orbit. Still, since the inclination of Saturn's axis (and hence its rings) is about 27 degrees, this is impossible. But unfortunately, there is no Simply Switch to help here. If Saturn were spherical, all orbits around its center would be equivalent. However, it has an editorial convexity that tends to pull particles into its plane.

Edited July 15, 2021 by basingset