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Hill Sphere period - another formula (Read 23048 times)
frankuitaalst
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Hill Sphere period - another formula
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!.
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abyssoft
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Re: Hill Sphere period - another formula
Reply #1 - 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.
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #2 - 09/06/08 at 14:57:38
 
Quote from 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.

I'm curious where you get these formulas from or how they were derived ...
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Re: Hill Sphere period - another formula
Reply #3 - 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.  
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #4 - 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.
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Re: Hill Sphere period - another formula
Reply #5 - 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
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #6 - 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 ?
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #7 - 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 .
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Hillsphere.jpg
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Re: Hill Sphere period - another formula
Reply #8 - 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.
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Re: Hill Sphere period - another formula
Reply #9 - 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.
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #10 - 09/09/08 at 09:20:24
 
Quote from 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.
.  
Is there a problem with Vista running GravSim ( and other applications ?) . I still hesitate to convert from XP to Vista.  
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Re: Hill Sphere period - another formula
Reply #11 - 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.
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Re: Hill Sphere period - another formula
Reply #12 - 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
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Re: Hill Sphere period - another formula
Reply #13 - 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
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frankuitaalst
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Re: Hill Sphere period - another formula
Reply #14 - 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.
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JupiterHillSphereReverse.gif
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