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General >> Discussion >> New system - asteroid belts & kirkwood gaps pt. 2
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1223246512

Message started by Mal on 10/05/08 at 15:41:51

Title: New system - asteroid belts & kirkwood gaps pt. 2
Post by Mal on 10/05/08 at 15:41:51

Here's a brand new system I made in order to see what happens with kirkwood gaps within its asteroid belt. This is actually the one that started off my interest in it, because as it is it has a planet between the 3:2 and 2:1 resonances of the innermost jovian, and that probably shouldn't be 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
 
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
 
1. Orbit = 0.300 AU, Radius = 4000. km, Density = 5000. kg/m3, Mass= .001 MJ, BBTemp=  511.4 K, MMW=   28.35, Atm=   1
2. Orbit = 0.570 AU, Radius = 4800. km, Density = 3500. kg/m3, Mass= .001 MJ, BBTemp=  371.0 K, MMW=   20.40, Atm=  10
3. Orbit = 1.454 AU, Radius = 8000. km, Density = 3500. kg/m3, Mass= .004 MJ, BBTemp=  232.3 K, MMW=    4.60, Atm=  15


Planet 3 is the one that's between the 2:1 and 3:2 resonances... and also is on an eccentric orbit so it actually goes through the asteroid belt too. So it probably isn't really a stable system with this planet there (that said, over about 650 years it seemed stable...)

I did two runs with 200 asteroids located at 1.176 +/- 23% AU, with the other planets as shown above (I put all of them in). In the first run I left out the errant planet 3, and it seemed pretty stable. In the second run, I kept planet 3 in there and had a few asteroids tossed into more eccentric orbits. I ran both for about 500 years at 16k timestep (I did them originally at 65k, but that may have been too fast so I re-ran them at a slower one).

Frank - would you be able to do some sma/ecc graphs for the asteroids in this system again at all? I'd be really interested to see if (a) any kirkwood gaps are starting to show up (though that said, the belt is pretty much exactly fills the space between the 4:1 and 2:1 resonances of the inner jovian (Planet 5)) and (b) what effect Planet 3 has on the belt in the second run.

You can download the output files here (showing a and e), they're about 5MB each (zipped) - I'll leave these up for a couple of days:
Run1 (no planet 3): http://www.spicapublishing.co.uk/users/consty/2840nopanth.zip (16k timestep)
Run2 (with planet 3): http://www.spicapublishing.co.uk/users/consty/2840panth.zip (16k timestep)

Title: Re: New system - asteroid belts and kirkwood gaps
Post by Mal on 10/05/08 at 19:52:57

Here's a picture of the second run (with the extra planet in blue) when it starts. As you can see, it goes into the belt (the belt is between 0.909 and 1.444 AU, and the planet's perihelion is 1.217 AU).

Title: Re: New system - asteroid belts and kirkwood gaps
Post by Mal on 10/05/08 at 21:44:04

This is the 2nd run with the blue planet, after 510 years at 16k timestep.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/06/08 at 00:52:05

And here's the first run (with no blue planet) after about 531 years. As you can see, it's a lot more stable, and the belt is hardly ruffled at all.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/09/08 at 00:24:42

I'm trying another run with 250 asteroids at 1.3 AU +/- 50% at 16k (no blue planet). The first run didn't really show much because all the asteroids were between 2:1 and 4:1, and all they did was jiggle up and down a bit on the a-e graph (Frank's let me use his graph plotter, which is proving to be interesting!). This new run should have asteroids that go from just inside 4:1 to beyond the 3:2 distance so I should see a more interesting evolution (they go pretty much to the orbit of the second planet, out to just within the innermost jovian).

I'll leave it running overnight and I'll try and post some results tomorrow night if I can.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/09/08 at 18:26:44

Here's an sma-ecc plot of the 250 asteroids after about 2800 years at 16k timestep. You can't see it very well but there's a definite disruption around 1.44 AU (the 2:1 resonance with the innermost jovian) and beyond that it's pretty chaotic, but the weird thing is that the entire belt is becoming more eccentric as time goes on as you go further from the star (to the right on the graph - the lefthand blue line is at 0.5 AU, the righthand blue line is at 2.3 AU with 0.2 AU increments in between). The belt started off at 0 ecc, and after 2800 years the main belt is peaking around 0.05 ecc. I wonder if that's because of the other jovians in the system, or if something else is afoot - any ideas as to what could be going on here?

I'm not really seeing that eccentricity pumping/surging that was going on in our own solar system either, but I'm not sure if that's because I've got too few asteroids plotted here (maybe I need a few hundred more too see that?).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/10/08 at 10:52:10

Mal , Is this a picture with or without the extra planet ?

EDIT : is the picture a representation of the files you provided above , or is it a different fileoutput ?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/10/08 at 13:47:02

it's without the extra planet (I've added labels to the graph, you may have to refresh your browser to see them).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/10/08 at 14:20:43

If the picture above is from the file you provided above , the animation looks as following .
I think it's normal the eccentricity rises as the  bodies are further from the central sun.
If you look closely , if possible ,  the particles move initially in a kind of wave .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/10/08 at 14:27:06

Hm I'll have to try that again... the picture isn't from the text files in my first post - it's from a new run of 250 asteroids at 16k for 2800 years that I mentioned a few posts up. I've attached the gsim file.

(how are you making the animationw btw, is that just in an animated gif creator?)

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/10/08 at 14:35:43


Mal wrote:
(how are you making the animationw btw, is that just in an animated gif creator?)

Yes indeed , it takes three programs , first GravSim , then the SMAEccInclViewer and finally a gif creator .
( Whished all could be done in one program   :))  

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/10/08 at 19:30:35

Yeah I ran the original run again for 150 years and got the same results as Frank... completely wacky, what the heck is causing the eccentricity to do that??? It's like there's a wave propagating through it from the sunward end whose frequency increases over time.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/11/08 at 02:16:39


Mal wrote:
Yeah I ran the original run again for 150 years and got the same results as Frank... completely wacky, what the heck is causing the eccentricity to do that??? It's like there's a wave propagating through it from the sunward end whose frequency increases over time.

I'm glad the programm also runs at your PC .
Resonances are quite complicated , that's why I wrote this visualisation tool to get some feeling with it . In fact you can compare it with a string of spring masses on which you tug with a force with a given frequency . You'll get also a wave pattern .
In this case the force however acts in two dimensions and with a 1/rČ scale , creating this weird patterns .
Maybe you'll find some principles in the work of Murray & Dermott . I haven't looked up yet .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Tony on 10/11/08 at 14:58:03


frankuitaalst wrote:
( Whished all could be done in one program   :))  

Me too.  But I've never found a GIF writer for Visual Basic.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/11/08 at 23:43:43

OK, I've kinda figured out how to make the graphs and animate them (thanks to Frank!)...

Here's about 150 years worth of 200 asteroids at 1.176 +/- 23% AU and 16k timestep. This first file is without the blue planet, so the system is as shown below:


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
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


Here's where the resonances are for the inner jovian (at 2.291 AU)


Code:
Res     AU
4:1     0.91
3:1     1.10
5:2     1.24
7:3     1.30
2:1     1.44
5:3     1.63
3:2     1.75
4:3     1.89
5:4     1.97
1:1     2.29


(the blue lines are 0.06 AU apart, starting at 0.9 AU on the left).

The big bulge forming on the right is the 2:1 resonance (which marks the outer boundary of this particular belt). You can see a much smaller bump forming at 1.10 AU, which corresponds to the 3:1 resonance. You can see the 'flicking' starting on the left edge of the belt and working inward during the first few years, and very rapidly the whole belt is jiggling up and down in sma-ecc space.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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...

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by 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?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/12/08 at 00:27:00


frankuitaalst wrote:
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 :).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/12/08 at 01:20:10

STOP THE PRESSES!!! :)

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 ;) ), not caused by jovians or any other bodies.

Anyone got any ideas about this???

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/12/08 at 01:23:39


Mal wrote:
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 .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/12/08 at 01:38:09


Mal wrote:
STOP THE PRESSES!!! :)
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 .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 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 .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 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  :), 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!)

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/15/08 at 08:49:15

This is the same graph with a max eccentricity of 0.3... I've drawn a white line in - is it me or can you draw a line between the jovian and the deflected sunward part of the outer belt?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/16/08 at 09:04:17

This graph (using the new beta) shows:

An inner belt of 250 massless point asteroids (2.87 - 5.3 AU)
A jovian of 0.379 MJ, at 2.291 AU with an ecc of 0.1
An outer belt 250 massless point asteroids (0.58 - 1.61 AU).

I recalculated the inner and outer edges of the "gap" around the jovian based on its new eccentricity, which is why the extent of the belts isn't the same as before when it was 0.03.

I'm not sure why the inner edge of the outer belt is so ragged while the outer edge of the inner belt is so sharply defined. It could be that the calculated "outer reach" of the jovian is too low (I used the formula from this Jones/Sleep/Underwood paper (http://www.journals.uchicago.edu/doi/abs/10.1086/428108) which you can download by clicking the PDF Version link at the top of the page. TBH I'm surprised they're giving it away for free!).

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by abyssoft on 10/16/08 at 11:18:30

:D Ty for the wonderful link it will help me a lot with my sims

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 10/16/08 at 12:44:41

yah, it's a good paper for worldbuilders :).

I should see if I can get some orbital dynamics postdocs on here... I'd love to know what they think of some of this stuff.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/16/08 at 13:00:53


Mal wrote:
- just like the interior resonances closest to the jovian were angled inwards. Again, could this be down to the eccentricity of the jovian?

I think this must be the result of the resonances themselves .
I got the same result in other simulations with 0 eccentricity of the Jupiter planet .
See the post on Dynamics of Resonances .


Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 10/16/08 at 22:56:11


Mal wrote:
This is the same graph with a max eccentricity of 0.3... I've drawn a white line in - is it me or can you draw a line between the jovian and the deflected sunward part of the outer belt?

I think there's some parabolic relationnship between a and e of a body in resonance with the main body . The parabola bends inwards for inner bodies and outwards for outer bodies ( see the latest posts in the Dynamic of resonances thread for 1:2 and 2:1 resonances ) . Saying the kissing line goes through the main planet can hold , but depends on the place where you start the parabola .

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 11/02/08 at 15:33:22

Something a bit different here:

This is a graph showing 150 years of evolution of a belt of 500 massless point asteroids that start off linearly distributed between 3.0 and 5.0 AU within Jupiter's orbit in our own solar system (all other bodies have been removed except for Jupiter and the Sun). So the belt starts just within the 2:1 resonance and goes out almost to Jupiter's orbit at 5.204 AU.

As you can see, the results are rather... explosive - the belt is quickly torn apart and asteroids are sent scattering all over the place. I think this confirms that it's definitely impossible for a planet to form within this region close to a jovian - all the planetesimals would be in orbits that are too unstable (and if I left this running long enough I suspect that the belt would dissipate eventually).

The big peak on the left is the 2:1 resonance. I think the "wall" to the right of that (beyond which everything goes crazy) is actually roughly where the 3:2 resonance is.

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by Mal on 11/03/08 at 17:57:27

The SMA/Ecc graphs are pretty good for showing how the eccentricity gets pumped, but I'm still not sure where there are actual gaps in the belt (or maybe 150-200 years is too short a timescale for that to happen). Right now it looks like things are keeping the same SMA but increasing their eccentricity... What would be the best way to find out if the SMA itself is changing?

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 11/05/08 at 11:31:42


Mal wrote:
The SMA/Ecc graphs are pretty good for showing how the eccentricity gets pumped, but I'm still not sure where there are actual gaps in the belt (or maybe 150-200 years is too short a timescale for that to happen). Right now it looks like things are keeping the same SMA but increasing their eccentricity... What would be the best way to find out if the SMA itself is changing?

In this case I think you can use the time feature which allows you to plot the SMA vs time. The viewer also outputs the MeanSMA as a time series . This might be helpful ...

Title: Re: New system - asteroid belts & kirkwood gaps pt
Post by frankuitaalst on 11/05/08 at 13:01:16


Mal wrote:
Something a bit different here:
As you can see, the results are rather... explosive - the belt is quickly torn apart and asteroids are sent scattering all over the place. I think this confirms that it's definitely impossible for a planet to form within this region close to a jovian - all the planetesimals would be in orbits that are too unstable (and if I left this running long enough I suspect that the belt would dissipate eventually).

The big peak on the left is the 2:1 resonance. I think the "wall" to the right of that (beyond which everything goes crazy) is actually roughly where the 3:2 resonance is.

Amazing plot Mal .
Jupiter has a strong disturbing effect . I think you might get a simular result if you put the bodies between Jupiter and Saturn

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