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 Hill Sphere period - another formula (Read 34666 times)
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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!. Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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. Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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 ... Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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. Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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. Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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 Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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 ? Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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 . Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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. Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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. Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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. Back to top IP Logged
 Tony YaBB Administrator Posts: 1060 Gender: 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. Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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 Back to top IP Logged
 abyssoft YaBB Administrator I love YaBB 1G - SP1! Posts: 302 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 Back to top IP Logged
 frankuitaalst Ultimate Member Great site Posts: 1510 Gender: 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. Back to top IP Logged
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