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New system - asteroid belts & kirkwood gaps pt. 2 (Read 18514 times)
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #15 - 10/11/08 at 23:48:02
 
For comparison, here's the same system with the same asteroids, but with the extra (blue) planet:
 
Code:
   Num Star	Age    ZMass   Mass	  ZLum	     Lum	   Rad   Met Pha
  2840 G2 V     4.736   0.992   0.992     0.7072248     1.0207404     0.997   2   1
  
ZAMS Inner Zone =    0.704   Current Inner Zone =    0.833
ZAMS Hab Zone   =    1.407   Current Hab Zone   =    1.668
ZAMS Earth Equiv=    0.841   Current Earth Equiv=    1.010
ZAMS Outer Zone =    2.132   Current Outer Zone =    2.562
 
 Habitable
Num    AU	  ecc	 type	mass     den	 rad	  peri	 aph	hill     inner     outer	4:1	 2:1	migrate
 1     0.300     0.044     4.000     0.001  5000.000  4000.000     0.287     0.313     0.002     0.279     0.319     0.119     0.189     0.671
 2     0.570     0.094     4.000     0.001  3500.000  4800.000     0.516     0.624     0.004     0.498     0.635     0.226     0.359     0.671
 3     0.969     0.000     5.000	Planetoid Belt
 4     1.454     0.162     4.000     0.004  3500.000  8000.000     1.217     1.690     0.016     1.107     1.729     0.577     0.916     0.671
 5     2.291     0.030     2.000     0.379  1246.700 51658.500     2.223     2.358     0.113     1.769     2.699     0.909     1.443     0.671
 6     4.123     0.063     2.000     0.350   993.500 54227.500     3.863     4.383     0.199     2.868     4.980     1.636     2.597     0.671
 7     7.834     0.036     3.000     2.123  2340.200 74357.000     7.554     8.113     0.689     4.797    10.181     3.109     4.935     0.671
 8    16.451     0.044     1.000     0.236  1700.280 39746.000    15.726    17.175     0.696    12.944    19.261     6.528    10.363     0.671
 9    24.676     0.025     1.000     0.196  1706.880 37318.000    24.048    25.304     0.981    20.124    28.246     9.792    15.545     0.671 


 
You can see the blue planet at 1.454 AU at the top right of the graph - it moves up and to the left as time goes on, but it's coming back down to its original position on the graph at the end. As you can see, its influence makes the outer half of the belt rather chaotic as it kicks lots of asteroids up to more eccentric orbits. On this short timescale it's not as unstable as I thought it would be, but I suspect if I left it running for a few million years then the blue planet would be booted out of the system and the asteroid belt would be a lot thinner...
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #16 - 10/12/08 at 00:07:58
 
Hi Mal , do you use a special program in order to set up a system in search for habit zones?
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #17 - 10/12/08 at 00:22:13
 
Something a bit different now:
- I've removed all the planets except for the innermost jovian, just to rule out any effects from any other bodies.  
- I've now got 250 asteroids between 0.65 and 1.95 AU (1.3 AU +/- 50%). This means the belt is much wider than before, and fully includes the 2:1 and 3:2 resonances so we can see what's going on there.
 
Code:
   Num Star	Age    ZMass   Mass	  ZLum	     Lum	   Rad   Met Pha
  2840 G2 V     4.736   0.992   0.992     0.7072248     1.0207404     0.997   2   1

Num    AU	  ecc	 type	mass     den	 rad	  peri	 aph	hill     inner     outer	4:1	 2:1	migrate
 3     0.65-1.95 AU	 Planetoid Belt
 5     2.291     0.030     2.000     0.379  1246.700 51658.500     2.223     2.358     0.113     1.769     2.699     0.909     1.443     0.671 


 
 
The blue lines on the graph below are 0.15 AU apart - it starts at 0.5 AU on the left and extends to 2.0 AU on the right.
 
You can see several things here:  
 
- the 'flicking' from the left edge of the belt is still there. Interestingly the amplitude of the flicking is largest closest to the sun, and decreases toward the centre of the belt. The innermost asteroids aren't having their eccentricities bumped up here (this was happening in the other two animations) - presumably this must have been caused by the planets inside the belt that aren't there in this run.  
- the 2:1 resonance at 1.44 AU is really obvious, causing a big peak in the eccentricities.  
- the 3:1 resonance at 1.10 AU is missing because that's exactly where that gap in the asteroid belt distribution fell that I mentioned in my 'non-uniform asteroids' thread. Presumably it would be visible as a smaller bump otherwise...  
- the 5:3 resonance at 1.63 AU is visible as a smaller 'spike' in the graph, to the right of the 2:1 bump.  
- the 3:2 resonance at 1.75 AU sends a few asteroids shooting up the graph to the right of the 5:3 spike. It's kinda hard to see because beyond that you're getting into the region where asteroids get chaotically jumbled by overlapping resonances (as shown on p458 in SDD).  
 
Left for long enough, I think it's quite likely that the resonances would become gaps or clumps like in our own belt so I think this proves that belts in other systems would have their own kirkwood gaps.
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #18 - 10/12/08 at 00:27:00
 
Quote from frankuitaalst on 10/12/08 at 00:07:58:
Hi Mal , do you use a special program in order to set up a system in search for habit zones?

 
I've got my own Fortran program that generates systems (it generated this one). This particular system is one that supposedly had a habitable planet in it... but I picked it out here because the rocky planet shouldn't actually be there - it's the blue planet that (a) goes through the belt and (b) is near the 3:2 resonance with the jovian but beyond its "inner limit" within which the rocky planet would be ejected from the system. Chances are that the planet wouldn't have been able to form in the first place given its location, so this is actually a buggy system Smiley.
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #19 - 10/12/08 at 01:20:10
 
STOP THE PRESSES!!! Smiley
 
I had a suspicion about this, so I tried it out - I removed ALL the planets and just left the belt to evolve on its own for 160 years, just to see what happened. As you can see from the animation below, even with no other bodies in the system (apart from the sun) the 'flicking' that propagates outward from the inner edge of the belt is still there!! Though obviously since there's no jovian anymore, the bumps at the resonances aren't there anymore.  
 
The only possible reason I can think of for this (unless this is some bug in Gravity Simulator itself or something to do with the timestep?) is that it's because the asteroids aren't massless - when I defined them I gave them a radius of 0.5 km and a mass of 100,000,000 kg (100,000 metric tons). So could they be perturbing eachother, and that's what's causing the flicking? I also wonder if the reason that the 'flicking' starts at the inner edge of the belt is because those asteroids are the ones that are orbiting the fastest - as the outer asteroids complete their orbits up the 'wave' propagates outward too? Though I still don't know what the flicking actually is.... Either way, I really can't think of any other cause for the flicking given that there's no other bodies here - I guess the only way to find out for sure would be to recreate the belt but make the asteroids massless points (can we do that in GravSim?) and see what happens...  
 
But at least now we seem to have proven that the 'flicking' is a natural property of the belt itself (or a bug in GravSim or a timestep issue Wink ), not caused by jovians or any other bodies.
 
Anyone got any ideas about this???
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #20 - 10/12/08 at 01:23:39
 
Quote from Mal on 10/12/08 at 00:22:13:

Left for long enough, I think it's quite likely that the resonances would become gaps or clumps like in our own belt so I think this proves that belts in other systems would have their own kirkwood gaps.

I'm definitively sure about this .  
There's even a thought now to detect extrasolar planets which aren't visible or detectable , but whose presence can be assumed  as they may leave a gap in the dust around the central star .  
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #21 - 10/12/08 at 01:38:09
 
Quote from Mal on 10/12/08 at 01:20:10:
STOP THE PRESSES!!! Smiley
Anyone got any ideas about this???

I've an idea about this phenomenon , but I'll leave it to Tony to come with an explanation .  
For sure : no bug .  
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #22 - 10/12/08 at 02:27:07
 
Well I'm stumped. Here's a 60 year animation of the belt, but this time the asteroids have 0 mass and 0 radius, so they're massless points. And still the flicking is there (and it looks pretty much the same as the previous graph, so I doubt if it's anything to do with the asteroids perturbing eachother). To give an idea of scale, the eccentricity is varying between the order of 1e-5 and 1e-3 here.  
 
Given that it's still happening with massless points (so they can have no gravitational influence at all on eachother), could this just be down to some kind of rounding/calculation error in the program? I can't think of anything else it could be, since gravity can't be the cause.  
 
Though strangely enough, if you look at the animation, the flicking seems to go from left to right from the inner edge of the belt, but from right to left from the middle/far edge of the belt, and it seems to converge on 1 AU. Could that signify anything?
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #23 - 10/12/08 at 02:45:17
 
In my opnion this is due to the algoritm of integration of the Euler method .  
This method works very well , but induces some inaccurancies of the positions of the asteroids . The closer the objects come , ie as speed increases the error increases also .  
In the long term the effect is almost zero , but individual values tend to oscillate around the exact value .  
If you run the same at smaller timesteps the error should decrease also .  
I wonder how Tony interpretes this .
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #24 - 10/12/08 at 10:18:14
 
I've made a new thread for all this now, so I'll carry on with it over there.
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1223831545
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #25 - 10/13/08 at 10:32:20
 
Here's a cumulative picture of what happens if you have the 250 asteroids with just the inner jovian. You can more clearly see the resonances here - the first big spike (from the left) is the 2:1, the second one is the 5:3, and the next big one going up to the top is 3:2, and then there's a mess of orbits on the far right beyond 1.849 AU caused by the resonance overlap zone.  
 
One odd thing you can see clearly here is that the 5:3 and 3:2 spikes are angled inward - could that be because of the jovian's eccentricity? (0.03).
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #26 - 10/13/08 at 12:33:44
 
And here's what happens if you put the belt beyond the jovian. This is 250 asteroids at 5.4AU (+/- 50%, between 2.7AU and 8.1 AU), to see what happens with the exterior resonances.
 
Here's the list of external resonances:
 
Code:
Res     AU
1:1     2.29
7:8     2.50
6:7     2.54
5:6     2.59
4:5     2.66
3:4     2.78
5:7     2.87
2:3     3.00
5:8     3.13
3:5     3.22
1:2     3.64
3:7     4.03
2:5     4.22
3:8     4.41
1:3     4.77
2:7     5.28
1:4     5.77
1:5     6.70
1:6     7.56
1:7     8.38
1:8     9.16 


 
I think the first big spike on the left is the 3:4, and the next big spike is the 2:3. The smaller spike after that looks like the 1:2, and there's possibly a tiny peak at about 4.8 AU that might be the 1:3. Interestingly enough it seems from this that the lower order 3:4 and 2:3 resonances are more important than the 1:2s and 1:3s (unlike the internal resonances, where the X:1 resonances are more important). I'll also note that the 3:4 and 2:3 resonances look like they're angled outwards a bit too (the 2:1, not so much) - just like the interior resonances closest to the jovian were angled inwards. Again, could this be down to the eccentricity of the jovian?
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« Last Edit: 10/13/08 at 14:19:53 by EDG »  

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Re: New system - asteroid belts & kirkwood gaps pt
Reply #27 - 10/13/08 at 15:39:41
 
Here's both an inner and outer belt at the same time. The inner belt is 250 massless point asteroids at 1.16 AU +/- 50% (0.58 - 1.74 AU), and the outer belt is 250 massless point asteroids at 4.0 AU +/- 32.5% (2.7 - 5.3 AU). So the outer belt is a bit more densely packed than the inner one. I've labelled where I think the resonances are.
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #28 - 10/14/08 at 01:00:45
 
Ok, this is a bit interesting - this is the same two-belt set up as the previous post, but with the jovian at zero eccentricity instead of 0.03. Both simulation ran for 60 years but look quite different. In the zero eccentriciy simulation you can clearly see the 2:1, 3:2, 4:5, 2:3, and 1:2 spikes... but the 3:1, 5:3, 3:4 and 3:5 spikes are no longer visible! Also, the interior resonance spikes look pretty straight, but the exterior resonance ones look like they're angled outward, even with the jovian at zero eccentricity - most odd!
 
(I didn't try this with the new beta because I couldn't get the belts set up properly. Now that Tony's corrected it I can try again in the next few days).
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Re: New system - asteroid belts & kirkwood gaps pt
Reply #29 - 10/15/08 at 00:32:08
 
Here's the results of a longer run with the eccentric jovian, done with linearly-distributed belts using the new beta. As you can see, (a) the new beta works fine now  Smiley, and (b) the linear function means that there's no annoying gaps in the distribution!  
 
So at the end of this 500 year run we can clearly see the inner resonances - from left to right the spikes are 3:1, 2:1, 5:3 and 3:2. In the outer belt, the sunward part is still pretty messy - within 3 AU we may have the 4:5 (but it's not too clear), and at 3 AU we have the 2:3 resonance, the solitary spike between 3 and 3.5 AU is the 3:5, the big spike outside 3.5 AU is the 1:2 (not sure why it bifurcates at the top though) and now we can see the 1:3 bump forming between 4.5 and 5 AU.  
 
It's interesting to see the speed at which these resonances start to affect the eccentricity... interior to the jovian the 2:1 and 3:2 develop the fastest, but outside it seems that the lower order resonances (4:5 and 2:3) develop quickest (but the 1:2 and 3:5 are still pretty big).  
 
Though I'll have to figure out a way to see if there are any actual 'kirkwood gaps' in the sma here - you can't really see them using this method of graphing.
 
EDIT: And having made an animation of this (which is way too big to post here right now), it's apparent that those spikes are largely caused by a handful of asteroids moving up and down (the 3:1 is caused only by a couple of bodies!)
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