Gravity Simulator
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl
General >> Discussion >> Hill Sphere period - another formula
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1220711839

Message started by frankuitaalst on 09/06/08 at 07:37:19

Title: Hill Sphere period - another formula
Post by frankuitaalst on 09/06/08 at 07:37:19

I've been wondering what the period can be of a planet or moon oriting at the hill sphere at distance r .
http://orbitsimulator.com/gravity/articles/hillsphere.html
gives the radius (or Sma) .
Amazingly the period t of the orbiting moon is easily derived as : t= T * sqrt ( (1-e)^3)/3).
or in case of a circular orbit of the planet as : t= T * sqrt (1/3) .
In case of the Sun-Earth-Moon system the t becomes : 1 year/1.73 or about 7 months!.

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/06/08 at 14:25:06

unfortunately the hill sphere is not a stable region for orbits

however HillSphere * (Phi/e)^Phi^(e/2) is stable for prograde orbits and HillSphere * (Phi/e)^Phi is stable for retrograde orbits.

where Phi is the golden number (sqrt(5)+1)/2 and e is the natural number.

I'll be posting my work on binary stellar bodies, binary planets, dwarfs, and sssbs; sometime in the next week or two.

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/06/08 at 14:57:38


abyssoft wrote:
unfortunately the hill sphere is not a stable region for orbits

however HillSphere * (Phi/e)^Phi^(e/2) is stable for prograde orbits and HillSphere * (Phi/e)^Phi is stable for retrograde orbits.

where Phi is the golden number (sqrt(5)+1)/2 and e is the natural number.

I'll be posting my work on binary stellar bodies, binary planets, dwarfs, and sssbs; sometime in the next week or two.

I'm curious where you get these formulas from or how they were derived ...

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/06/08 at 20:32:10

I used empirical data and then extrapolated from the data.

it has been know that the maximum extent of stability was between 0.33 and 0.5 x Hill Sphere

Using the Gas Giants on can see this range.  If you simulate the the Jovian system in it's entirety you will see that over geologic time the outermost retrograde moon is actually in an unstable orbit it is just beyond the retrograde stability zone

S/2003 J 2 see entry http://en.wikipedia.org/wiki/S/2003_J_2 from wikipedia


Quote:
Retrograde moons with axes up to 67% of Hill radius are believed to be stable.
 This statement from the article is disputable as there is not reference to it give in the article. and there is another article that better explains the Hill Sphere http://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability  Using simulations of n bodies, HillSphere * (Phi/e)^Phi was the maximum range That I could keep a retrograde orbit stable indefinitely.

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/07/08 at 01:27:53

Thanks for the information abyssoft.
Aside from your scepsis about the article I think also the influence of other planets such us Saturn may come into role ?
Time to set up a GravSim ?

BTW : the title above is not that good . It should rather be : ....- another representation.

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/08/08 at 20:52:25

Ok I'm not 100% done but I am ready for some feed back

My view on binaries from an analytical view point with applied mathematics.

Planetary Debate and Binaries
http://docs.google.com/View?docID=dm63xwh_29srm2rfdb&revision=_latest

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/09/08 at 01:20:01

I have a question about the phrase in the link above :
"Once the hill sphere has been determined next we need to determine the semi-major axis outer boundary for pro-grade orbits; this is done by taking the hill sphere * ( Phi / e ) ^ Phi ^ ( e / 2 ).  Just to insure we have all necessary data points we should also calculate the semi-major axis outer boundary for retrograde orbits; this is done by taking the hill sphere * ( Phi / e ) ^ Phi. "
I have known the Hill sphere as an upper limit of stable bound orbits , but I'm curious to see how te above formulae were derived ?

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/09/08 at 05:53:12

I think the article in Wiipedia mentioning a maximum stabilty region to be between 1/2 and 1/3 of the Hillradius is acceptable .  I've checked wit GravitySimulator the stability for the Sun-Jupiter system doing the following :
Created the Sun-Jupiter system , added 300 zero masses at 0.15AU+/-50% around Jupiter and let run for ca. 40 years , at a small time-step.
A lot of  exit the system , but about 20% stay stable.
The result of the output in Excel ( after deleting the instable ones ) shows the variation of the SMA of the bodies which can be accepted as being "stable".
The more closer they start the less variation in SMA occurs.
Looking at the picture I guess one can attribute a maximum SMA of about 22.000.000 km for this system , or perhaps a little bit more .  
This corresponds with 0,41 rHill.

BTW : running this system makes really fun. One can see the ring of bodies becoming elleptical , keeping a circular "bulge" . Instable ones leave the system creating "arms" . Looks like a small galaxy.
If wanted i can provide the .gsim .

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/09/08 at 06:58:09

I derived them empirically by plotting out the data and performing curve fitting the best curve fit I could achieve where those that are the result listed in the paper.  It was not through any advanced mathematical process.

the outer boundry for prograde results in a constant value close to 0.319535834655967
the outer boundry for retrograde results in a constant value close to 0.431962327235041

one could if one wanted to replace ( Phi / e ) ^ Phi ^ ( e / 2 ) with 0.319535834655967 and ( Phi / e ) ^ Phi with 0.431962327235041

note that my curve fit does not take in to account any other force other then gravity.

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/09/08 at 07:01:21

and I now have bad news my older XP box has now died and I can no longer run simulations as I an unable to get GravSim to run stably on my vista box.

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/09/08 at 09:20:24


abyssoft wrote:
and I now have bad news my older XP box has now died and I can no longer run simulations as I an unable to get GravSim to run stably on my vista box.
.
Is there a problem with Vista running GravSim ( and other applications ?) . I still hesitate to convert from XP to Vista.

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/09/08 at 13:16:53

What kind of Vista problems are you having?  This is the first report of Gravity Simulator / Vista problem I've received.

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/09/08 at 15:59:54

I can only get the old any I mean old version to run at all

the last beta copy I had always locked up.

maybe if I get a new install I'll try again

Title: Re: Hill Sphere period - another formula
Post by abyssoft on 09/09/08 at 18:43:59

ok found a way to make sure it works correctly.

uninstall,

redownload, Right click and goto properties, unblock file, run as admin to install.

redownload all of the latest beta, again right click and goto the properties of each and unblock file.

then run. it then works correctly

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/10/08 at 01:36:01

Here's an animation of a gsim simulation about the Hill Sphere .
Around Jupiter (in the center) are orbiting 300  1kg masses in retrograde orbit . Orbits are initially circular , at random positionned at  0.2 AU +0.2 .
One sees the system quickly evolve towards an elliptical system and even to a system with spiral arms at opposite sides of the central Jupiter .
At first I wondered why the spiral arms are created at both sides , but then I realised this is due to the "tidal" effects of the sun. This is at first glance contra-intuitive .
I have the impression that a retrograde system allows much more eccentricity in the orbits than a prograde system .

BTW ; is there any way to attach also the .gsim file in the same post ? The system doesn't seem to accept multiple attachments.

Title: Re: Hill Sphere period - another formula
Post by Mal on 09/10/08 at 08:27:42

What would that look like with prograde orbits? (a mess, I'd imagine, since apparently those aren't stable near the hill sphere!).

I do like the spiral arms effect there...

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/10/08 at 08:58:11


Mal wrote:
What would that look like with prograde orbits? (a mess, I'd imagine, since apparently those aren't stable near the hill sphere!).

I do like the spiral arms effect there...

Right ! that's why I haven't posted them . Visuals should be kept somehow attractive  ;)
Well concerning the effect of the arms : it's even nicer to see how they form at simulation. The frames taken 1 year apart cover a lot .

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/11/08 at 16:02:08

An interesting thing about the objects that are right at the edge of stability, is that they can complete several orbits before being stripped away by the sun.  When they escape the planet, they always do so through the L1 or L2 points.  Then after orbiting the sun a few times, they get re-captured through the L1 or L2 point, complete a few orbits and then get ejected again.  It's fun to watch in the rotating frame as the positions of L1 and L2 are easy to spot.  Try this yourself:

File > New
Objects > Create -- Create a 1 Jupiter-mass object at SMA 1 AU.  Name it Jupiter.  Leave all inputs at their default.

Zoom out so you can see both objects.
View > Rotating Frame Adjustment.  Set it to the period of Jupiter.
Choose Rotating Frame.

Choose Jupiter as your focus object.

Create objects around it near the edge of stability:

Objects > Create Objects
With Jupiter set as the reference object, choose 0.31 AU as the SMA.  Change its color.  Leave all other inputs at their default.  Press Create.
Repeat, creating objects with sma's of 0.32, 0.33, 0.34, 0.35.  Play around with other values too.

Under Preferences, choose a graphics update interval of 50.

Set the time step to 512.

Sit back and watch as these objects escape Jupiter through the L1 and L2 points, orbit the Sun for a while, and get recaptured through the L1 and L2 points.

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/11/08 at 16:14:43

Nice remark ! I'll give it a try . How do you know this ?

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/11/08 at 18:16:15


frankuitaalst wrote:
...How do you know this ?

Because I know everything  ;D

Or just from trying different things.  Here's a screen shot of a system with a planet of 10 Jupiter masses orbiting at 1 AU.  A test particle is randomly placed in an orbit around the planet with an SMA of 0.8 AU.  It orbits a bit, escapes through L1, gets captured again through L1, over and over again.  This is very chaotic.  Any small change will ensure that you will not get the same results twice.

http://orbitsimulator.com/gravity/images/L1capture.GIF

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/11/08 at 18:18:44


Tony wrote:
[quote author=frankuitaalst link=1220711839/15#18 date=1221174883]...How do you know this ?

Because I know everything  ;D
http://orbitsimulator.com/gravity/images/L1capture.GIF[/quote]
That's what I thought  :)

Title: Re: Hill Sphere period - What is this?
Post by frankuitaalst on 09/12/08 at 03:02:38

Applying the input above as you mention Tony ( but instead of 1 Jovian - 10 Jovian masses ) at 1Au and 0.31 AU for the moon I got this amazing .gsim simulation .
My first reaction was " what the hell..." .
Thought first a had done something wrong . Simulation starts in rotating frame .
Till now my question is open .... :).
I'll post an animation later on .

Title: Re: Hill Sphere period
Post by frankuitaalst on 09/12/08 at 03:19:06

Here's the animation of the above gsim .
Screenshots every 0.1 years .
The body seems to travel between sun and Jupiter continiously .
I must have been lucky , because when I created the body it was exactly on the line between sun and jupiter .
Still don't know whats happening exactly , but I guess I must have created "by accident" a 0.5 resonance with appropriate eccentricity .

Edit : in order to investigate the parameters of the object I started the above simulation again, as I had saved it . Timestep 64 s.
To my surprise the object didn't have the linear orbit anymore but performed (also nice to see) a flowerlike orbit around the sun. Don't know whats happening or happens . Any idea Tony ?  I guess you other guys have the same result .

Animation:
http://www.orbitsimulator.com/yabbfiles/Attachments/WhatthehellzeropointoneyearsAnim.gif

Title: Hill Sphere period - Jupiter
Post by frankuitaalst on 09/12/08 at 12:22:05

I've tried to reproduce the simulation above , but no succes . I can't understand what happened and why opening the same sim afterwards gives a normal situation ...
Nevertheless , I performed the simulation Tony suggested , but this time with te real solarsystem . Put 100 bodies around Jupiter at 0.283 AU and let run .
Animation hereunder .
( actually I was anxious to see how the inner solar system would be affected ) .  
One can see the bodies escape over the L1 , other over the L2 point .
Note : I think the 0.283 AU setting was to much as the bodies escape rather quickly .

Animation:
http://www.orbitsimulator.com/yabbfiles/Attachments/100HillObjFromJupiter.gif

Title: Re: Hill Sphere period - What is this?
Post by frankuitaalst on 09/12/08 at 14:26:55

Concerning the mysterious linear motion above : I've been lucky to recover some more screenshots I made running this sim .
Screenshots start with Rotating frame to Jupiter , then I went to normal view , where the result is a elliptical orbit ( 1: 4 resonance to Jupiter I think) .
I can't understand how this elliptical orbit results in a straight line in rotating frame ...
( for the confort and "debugging" I made the animation extra slow)

Animation:
http://www.orbitsimulator.com/yabbfiles/Attachments/DetailLinear_motion.gif

Title: Re: Hill Sphere period - at the edge of stability
Post by frankuitaalst on 09/13/08 at 03:48:16

For the solarsystem a Hill radius of 0.155Au to Jupiter seems to be a configuration at the edge of stabilty ( in this orbital configuration starting at 2037) .
100 bodies were given 0.155 SMA around Jupiter
The simulation covers almost 100 years and consists of 400 individual frames.
One can see the system "wind up" for several tens of years , then some bodies escape to the inner solar system over the L1 point  one after another .
Watch how Jupiter seems to shoot every time it is near  aphelion . Only a few escape over L2 , leaving their orbit towards the outer solar system .
The animation appears to be of "poor" resolution or quality . This is due to the scaling and the reduction of colors and compression in order to fit the 1MB upload limit . The original amount of data before compression was about 1GB in bmp.

Animation:
http://www.orbitsimulator.com/yabbfiles/Attachments/100HillBodies0p155AU.gif

Title: Re: Hill Sphere period - Saturns stability
Post by frankuitaalst on 09/13/08 at 10:46:31

For comparison : Saturn allows about distance of 0.185 AU as stable region .
This  simulation shows Saturn from the point where it releases its first moons .
The orbits of the released moons seem to show less dynmices compared to these of jupiters moons.
Also the rate at which Saturn spews out the bodies is less compared to jupiter .

Animation:
http://www.orbitsimulator.com/yabbfiles/Attachments/Saturn100bodies0p185AU.gif

Title: Re: Hill Sphere period - Saturn
Post by frankuitaalst on 09/13/08 at 10:51:05

Prior to the release of the moons by Saturn ("wind up ") the system evolves from its initial configuration at 0.185 AU from saturn as shown hereunder .
Watch how the end of the animation a couple of moons escape at the left corner above over the L1 point .  

Animation:
http://www.orbitsimulator.com/gravity/images/users/frankuitaalst/Saturn0p185Detail.gif

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/13/08 at 12:04:01

Very neat animations!  As Jupiter gets closer to the sun, its Hill Sphere shrinks.

I edited the thread because having several large animated GIFs on one page causes the page to load too slowly, and makes the animations run slow as well.  So instead of seeing the animations, you see links to the animations which allow you to open them one at a time.

Saturn gets hidden off the bottom of the screen in one of the animations.  You can prevent this if you rotate the rotating frame with the > and < buttons.

There's a new feature in the latest beta (probably the one you're running now) in the view menu called Pointer.  So if you're zoomed in enough that you can't see the sun, you can at least have a pointer showing you the direction of the sun.  But there's a bug in this feature.  Sometimes the pointer's starting position is off the screen, which means you can't see it or drag it to a convienent place.  Try it anyway, as it might work for you.

Title: Re: Hill Sphere period
Post by frankuitaalst on 09/13/08 at 13:05:10


Tony wrote:
I edited the thread because having several large animated GIFs on one page causes the page to load too slowly, and makes the animations run slow as well.  So instead of seeing the animations, you see links to the animations which allow you to open them one at a time.

Well done so ...you're right , that was my point some time ago .
Can we as users also add some command to hide the annexe ?

BTW ; do you have any idea what was happening in the sim were there was a straight line in the rotating frame ?
Seems so weird , looks as if SMA and velocity are coupled in a strange ...way . Body must orbit a spiralling ellipse but I don't understand the physics , or might it be some bug ?

EDIT : if I try to open one of the attachments I don't get them . I get the message :" page not found" .

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/13/08 at 13:23:16

I'm getting that error too.

On my Admin control I have an option to delete the attachment.  I checked to see if "delete" simply meant "remove from post" or "delete image from server".  It seemed to only remove it from the post, as all the links worked when I previewed my changes.  But now they seem gone from the server too.  I wonder if it simply delays deleting it from the server.  I backed up one of the images, so I resored Saturn0p185Detail.gif but I think the others are gone.  I hope you still have them.  If so, you can e-mail them to me, and I'll upload them to a different folder, or you can re-attach them, and I'll modify them, but this time without deleting them.

The straight line one was weird.  You can't seem to recreate the simulation?  If you can, post the .gsim.

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/13/08 at 14:53:25

I'll reattach them the next days.

The weird .gsim : no I can't reproduce it .
What I did was the following : created a body around Jupiter , let it run , immediatelly in rotating frame , saw the weird line , let it run for some time , then in order to look at its real orbit switched to non rotating , let it run , and finally saved it as...
Opening the file to look at it again : the weird line was gone ....

Title: Re: Hill Sphere period - at the edge of stability
Post by Mal on 09/14/08 at 09:37:22


frankuitaalst wrote:
For the solarsystem a Hill radius of 0.155Au to Jupiter seems to be a configuration at the edge of stabilty ( in this orbital configuration starting at 2037) .
100 bodies were given 0.155 SMA around Jupiter
The simulation covers almost 100 years and consists of 400 individual frames.
One can see the system "wind up" for several tens of years , then some bodies escape to the inner solar system over the L1 point  one after another .
Watch how Jupiter seems to shoot every time it is near  aphelion . Only a few escape over L2 , leaving their orbit towards the outer solar system .
The animation appears to be of "poor" resolution or quality . This is due to the scaling and the reduction of colors and compression in order to fit the 1MB upload limit . The original amount of data before compression was about 1GB in bmp.


I'm wondering if I'm seeing things here. Follow the path that the particles make after they've left the hill sphere - is it me or does it look like they're in a triangular orbit? Isn't that usually seen with a 3:1 resonance?

Title: Re: Hill Sphere period - resonance ?
Post by frankuitaalst on 09/14/08 at 09:57:26

I had the same impression too , but the orbits are quite dynamic , haven't checked them already ...
But they ressemble indeed a 1:3 resonance ...

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/14/08 at 12:34:39

Some of them do get close to a 3:1 resonance, but not exactly, so the triangle drifts a lot from one resonant period to the next.  It's probably impossible for something to escape the planet and go into a resonance.

Title: Re: Hill Sphere period - SMA
Post by frankuitaalst on 09/16/08 at 01:36:43

I've a question about output :
The objects in the Hill sphere were created referenced to Jupiter . Later they will orbit the sun .
In the output values for SMA,e,...they appear still referenced to Jupiter .
Can one change this values so the output become related to sun?

Title: Re: Hill Sphere period - another formula
Post by Tony on 09/16/08 at 08:42:30

Objects > Edit objects
Change reference object

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/17/08 at 01:09:55


Tony wrote:
An interesting thing about the objects that are right at the edge of stability, is that they can complete several orbits before being stripped away by the sun.  When they escape the planet, they always do so through the L1 or L2 points.  Then after orbiting the sun a few times, they get re-captured through the L1 or L2 point, complete a few orbits and then get ejected again.  It's fun to watch in the rotating frame as the positions of L1 and L2 are easy to spot.  
Sit back and watch as these objects escape Jupiter through the L1 and L2 points, orbit the Sun for a while, and get recaptured through the L1 and L2 points.
.
I thought I remembered such a transfer , finally I found some reference in the thesis of Shane Ross :  
"Cylindrical Manifolds and Tube Dynamics in the RestrictedThree-Body ProblemThesis by
Shane David Ross"
Here's a representation from it's work . It is highly mathematical .

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/17/08 at 06:32:57

By chance I discovered in the work of Ross the orbit of Comet Oterma.
The orbit is really so special that I'll post it in a separate discussion ...

Title: Re: Hill Sphere period - evolution of orbital ..
Post by frankuitaalst on 09/26/08 at 12:04:13

In one of the above posts I posted an animation of 100 bodies around Jupiter , which are on the edge of stability at 0.155 AU from the planet . Some of them escape after a while , others remain bounded .

Hereunder is a dynamical plot of the evolution of the SMA and Eccentricity of each body in time .
Values were outputted every 0.04 years ....Bodies start in a circular otbit around Jupiter , close to 0.155AU
In the beginning the bodies spread out in Au , gain eccentricity and after a while show a whirling pattern ...
Enjoy the choreography in the beginning of the animation !

Title: Re: Hill Sphere period - another formula
Post by Mal on 09/26/08 at 20:26:34

These plots are awesome. You get so much more information by seeing how the system evolves through time. Plus, it's amazing how all these patterns and shapes just fall out of the physics :)

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/26/08 at 23:01:15

Yes , I find them fascinating too , thats the reason for posting  them :)
A general feature of the dynamics is the fact that systems evolve towards a chaotic stability out of a pure symmetrical initial state .
( It would be quite sensational to run the simulation backwards ! ) .
I whished I could provide the code or executable to generate these plots here on the forum , but the program still needs some manual input in order to set the scales before compiling .

Title: Re: Hill Sphere period - another formula
Post by frankuitaalst on 09/27/08 at 00:00:32

I've run the following sim again : it contains 300 bodies at 0.155AU around Jupiter , and outputted the a, e, i values every 5 days .  I used 256 as timestep .
Doing so the orbital parameters' evolution  in time can be represented in much more detail.

PS : its nice to see how the initial circular orbits change over time when running the gsim .  

Title: Re: Hill Sphere period - Evolution of Jup 0.155Au
Post by frankuitaalst on 09/27/08 at 00:05:02

The result of the processing of the SMA and Eccentricity of above simulation gives the following animation :
The system was simulated till the configuration looks as the first picture .

Advantage of this small timestep of 5 days is that the evolution of individual bodies becomes clear as one can follow the bodies individually .  

Gravity Simulator » Powered by YaBB 2.1!
YaBB © 2000-2005. All Rights Reserved.