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Hill region in a double star system (Read 9354 times)
frankuitaalst
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Hill region in a double star system
03/13/08 at 13:36:15
 
Hills region describes the maximal orbit a moon can orbit a planet .  
The normal formulas are only valid for small m/M ratios ( m being the planets resp. solar mass) .  
What in case of a double star system ? How far away can a planet orbit the sun in a stable orbit ?  
The following animation gives a hint :  
Simulated is a double star system , each sun's mass being Msun , orbiting each other at 5 AU .  
Added were 100 equidistant minor bodies orbiting the central sun ...
Frames were taken every 10 years . The simulation spans about 400 years.  
Result is that most "far" away bodies are ejected , those close to the sun stay in orbit , while those being "in the middle " switch orbits around each sun and get ejected after some time .  
The system seems to stabilize after about 350 years.
Out of this simulation the Hills radius can be estimated being < 0.5 DistSecondSun.
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DoubleSunAnim.gif
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frankuitaalst
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Re: Hill region in a double star system
Reply #1 - 03/13/08 at 13:39:26
 
Repeating the same simulation but putting the planets initially a little bit more outward gives the following result ...
Less planets stay in orbit around their sun . Again the maximum stable orbit is less than 0.5 Solar Distances .
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Tony
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Re: Hill region in a double star system
Reply #2 - 03/13/08 at 15:58:50
 
As I understand it, the Hill Sphere is the maximum distance anything can be and still complete 1 orbit.  For example, for Earth the Hill Sphere is about 1.5 million km.  But if you created an object at 1.4 million kilometers, it would still probably get ejected.  It might complete 1 or 2 orbits first.
 
Retrograde orbits are the most stable.  If you compute the Hill Sphere of each star in your double star system, using their point of closest approach (assuming an elliptical orbit), and you put retrograde planets around them (inclination: 180 +-0%), you should see a lot of objects orbiting in the outer fringes of the Hill Sphere.
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Re: Hill region in a double star system
Reply #3 - 03/14/08 at 20:43:24
 
Two things:  
 
First, all the major moons in the solar system are found within 1/3 of the Hill Sphere distance - beyond that apparently things get progressively more unstable (I think you'll find the outer asteroidal moons of Jupiter and Saturn are further than 1/3) - in practice a good part of the sphere still isn't really available for stable (prograde) orbits.  
 
Second, check this paper out, it's got a really nice empirical formula in it for calculating the stable orbits.  
Long-Term Stability of Planets in Binary Systems (Holman, Matthew J.; Wiegert, Paul A.) The Astronomical Journal, Volume 117, Issue 1, pp. 621-628.
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frankuitaalst
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Re: Hill region in a double star system
Reply #4 - 03/15/08 at 03:42:26
 
Thanks for the link , I'll go trough it .  
Retrograde orbits are far more stable then prograde orbits and even offer  curious paths  Smiley.  
To illustrate this I created two simulations :  
First : 100 bodies around the primary star of 1Msun in prograde orbit .  
Second : 100 bodies around the primary star of 1Msun in retrograde  orbit .
( both sims used the same initial conditions as in the second simulation the second star at 5AU was given an opposite velocity (*) ) .  
Animation below shows the behaviour of the bodies when the system gets settled after a couple of hundred years.  
Screens were centered around the Barycenter of both stars .  
 
First part shows the prograde orbit . Bodies stay "close" to the parent star .  
Second part shows the retrograde orbit . Bodies can be splitted into those staying close to their parent and those getting further  away .  
Third part of the animation focusses on those orbiting far away from their parent star .  
They tend to orbit in a trifoil orbit going throug the barycenter of the star system  Shocked!
 
(*) Question for Tony : as I took the first sim and copied it to a second and modified here only the velocity of the second star , I would expect a circular orbit around the barycenter . Instead I got a small eccentricity . Any idea why ?  
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CompositionDoublesun.gif
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frankuitaalst
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Re: Hill region in a double star system
Reply #5 - 03/15/08 at 04:35:49
 
Isolating one body out of the system above and representing it centered on the barycenter gives this amazing trifoil orbit .  Huh
The system seems dynamically stable ie. planet is not ejected but tends to modify its orbit slightly .  
Time of each screenshot is 1 year .  
Suns are 5AU apart .  
Such orbits may not be an exception in this configuration , as there where about 5 bodies in this class of orbit out of the original 100 bodies .
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« Last Edit: 03/15/08 at 09:42:19 by frankuitaalst »  

TrifoilDoubleStar.gif
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frankuitaalst
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Re: Hill region in a double star system
Reply #6 - 03/15/08 at 06:29:02
 
for those who want to try : here's the sim showing 5 bodies in this orbit .
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Tony
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Re: Hill region in a double star system
Reply #7 - 03/15/08 at 13:24:50
 
Quote from frankuitaalst on 03/15/08 at 03:42:26:
(*) Question for Tony : as I took the first sim and copied it to a second and modified here only the velocity of the second star , I would expect a circular orbit around the barycenter . Instead I got a small eccentricity . Any idea why ?

I'm not sure why.  Were your planets massless?  If not, that might explain it.
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frankuitaalst
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Re: Hill region in a double star system
Reply #8 - 03/15/08 at 13:49:48
 
Yes , the planets had 1 kg mass . I will try to do this again to check once more .  
Edit : I inversed the velocity of the second star once more , and again I get the same odd eccentricity .  
All the minors had mass 1kg . The star had Sun mass.
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Tony
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Re: Hill region in a double star system
Reply #9 - 03/15/08 at 14:09:28
 
1 kg should not have effected your simulation (it's 30 orders of magnitude less than the Sun).  In Gravity Simulator, when you use menu Time > Time Backwards, that's all it does is change the signs of the velocities of all the objects.  Circular orbits stay circular, so I'm not sure why yours isn't.  Perhaps you could send me a "before" copy of the simulation, and I could try doing what you're doing.
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frankuitaalst
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Re: Hill region in a double star system
Reply #10 - 03/15/08 at 14:48:35
 
In annex the GSim file .  
I tried to inverse the velocity of the second star . Can you repeat what I want to do in the barycenter mode ?
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Re: Hill region in a double star system
Reply #11 - 03/15/08 at 15:04:50
 
It works fine for me.  At first, they're orbiting counter-clockwise in identical circular orbits.  Then I change the signs of the velocity for the 2nd object in your list, and they (the two stars) orbit clockwise in identical circular orbits.
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frankuitaalst
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Re: Hill region in a double star system
Reply #12 - 03/15/08 at 15:17:09
 
Thats funny ... I use the 20Jan2008Beta version and get the eccentric orbit  cry
Edit object : change signs of the velocity .  
Focus on first star  
Toggle A-F .  
Do I do something wrong ?  
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Re: Hill region in a double star system
Reply #13 - 03/15/08 at 16:04:20
 
I'm using the same version.  Here's my exact procedure:
 
  • Open your file.
  • Speed up time to 2048 so I can see some action.
  • Press AF on the Graphics Options to center the barycenter
  • The green objects are chaotically orbiting.  The red star and the purple star are tracing identical circular paths, overplotting each other over the course of 1 orbit.  They move counterclockwise.
  • Objects > Edit Objects
  • Choose 2nd object in the list:  "Object"
  • Change the signs of its x and y velocities
  • Observe that the red and purple stars now trace identical circular paths, overplotting each other over the course of 1 orbit.  But now they move clockwise.
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frankuitaalst
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Re: Hill region in a double star system
Reply #14 - 03/15/08 at 16:22:54
 
cry I did exactly now what you did above and again I get the odd eccentric orbit . For input in the screen i used "ok" now  instead of "apply" but the result is the same .  embarrassed
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