Welcome, Guest. Please Login.
Gravity Simulator
11/22/17 at 04:56:46
News: Registration for new users has been disabled to discourage spam. If you would like to join the forum please send me an email with your desired screen name to tony at gravitysimulator dot com.
Home Help Search Login


Pages: 1 
Send Topic Print
Suppose that .... (Read 20573 times)
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Suppose that ....
04/06/07 at 13:08:04
 
GravSim is well suited to perform simulations ....in the sense of ...suppose that ..
Suppose that at the "beginning" the sun is circled by a bunch of equidistant planets of equal mass with eccentricity 0 ...Pure symmetry at the beginning .What will happen ?  . Will they stay in their perfect circles , will they interact , collide or will they be ejected ?  
The answer is  : depends upon .  
 
The next picture is the result of 20 planets each having a mass of 10*Mass Earth , perfectly distributed between the sun and 2 AU . This means 200 Earth masses in a distance of 4 AU ! Thats's a lot .
The result is ... chaos in  the long time for the outer planets . The simulation stopped at about 500 years when one of the outer planets collided . The inner planets stay remarkable stable , due to their strong binding to the sun . The outer planets however prefer to go elliptic and even exchange orbits ...searching for stabilty, which they don't find .  
The picture shows the evolution of the SMA of the outer planets from the start of the simulation .  
It seems that the planets tend to form "couples" at the beginning , meaning the gaps between the couples are bigger than the gaps between the initial planets . This causes of course stong interactions between the couples, leading to instability . Planets also may interchange orbits as can be seen in Planet14 and 12 ! .  
Stability definitevly depends upon the mass of the planets . Starting with planets of Earth mass gives a lot more stability in the system ...
 
  
 
Back to top
 

SMA_planets.jpg
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #1 - 05/14/07 at 14:09:55
 
Suppose that ...the solar system consists of the sun and 4 earths .  
Each of the 4 planets has equal mass ( Earth mass) , they all are at 1 AU from the sun , orbit perfect in a circle , but are all 90° shifted from each other .  
Perfect geometry .  
What will happen after a "while" ?  
Will they keep into orbit ? Will they collide ? Which one will collide ?  
Not easy to answer . I think the perfect configuration can hold on for a while , as the system is in perfect equilibrium . After some time some small perturbations have to build op to ...??
In order to create some "chaos" , ie equivalent to a real system , in the following sim the first planet was given an offset of only 5000 meters in the y direction . The others were at 90°, 180° and 270°.  
The system is steady for about 500.000 days , then the initial offset results in mutual attractions out of the steady state as can be seen from the picture . Looking from "above" all the planets remain for thousands of years within a small gap , as they were before . However the planets must come somehow in resonance as can be seen .  
I omitted the 3th  and 4th planet in order to keep the plot clear , but it seems to me a system of horseshoe orbits is created.  
Must be fun to see this in a rotating frame .  
Out of the plot one might think there is a certain cycle of about 10000 days ( some 30 years ) .
Back to top
 

4Earths90degrees.gif
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Suppose that ....
Reply #2 - 05/14/07 at 17:30:27
 
If the planets were at the lagrange points (say, L4, L5, and L3 relative to one of the planets) then one would expect some stability but I would have thought that the configuration you set up with them separated by 90° wouldn't have been anywhere near that stable...
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #3 - 05/14/07 at 17:46:45
 
This is similar to a discussion I'm taking part in on the BAUT forum:  http://www.bautforum.com/showthread.php?t=58590
 
Appearently, you can have trojan planets like Mal pointed out as long as their combined mass < about 4% the mass of the star they orbit.  I ran some sims with 2 planets spaced 60 degrees apart that remain intact indefinately.
 
I'm guessing that 3 planets would also be stable, spaced 60 degrees apart.  But how about 4 or more?  I'm going to guess that 4 and 5 are unstable, and 6 is stable, provided the 60 degree seperation.
Back to top
 
 
Email View Profile WWW   IP Logged
abyssoft
YaBB Administrator
*****


I love YaBB 1G -
SP1!

Posts: 302
Re: Suppose that ....
Reply #4 - 05/14/07 at 22:25:21
 
Tony Since you are a master of sim building I have one that relates to this thread I would like to see.
 
Star =1.45 Msol
6 planets=0.5Mjup
Placement of planets such that  
 
#1 @ 2.6 AU Base planet
 
#2 @ LG1 of Star and #1
#3 @ LG2 of Star and #1
#4 @ LG3 of Star and #1  
#5 @ LG4 of Star and #1
#6 @ LG5 of Star and #1
 
All Should be in 1:1 resonance with #1
Just to see what happens.
Back to top
 
 
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #5 - 05/14/07 at 22:47:55
 
your link to the baut forum provides a nice link indeed http://www.physics.montana.edu/faculty/cornish/lagrange.pdf for the situation of the lagrange points , where the mass limit is calculated for a 3-planet system at the L-points .  
I think however that there are solutions also for the 4 planet system as described above . One may think that the 4 planet systems evolves to a 3 planet system (without collision) so that 2 planets oscillate around one of the points .  
Similar as if they played "accordeon". This is just a guess .
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #6 - 05/15/07 at 02:55:41
 
Quote from abyssoft on 05/14/07 at 22:25:21:
Tony Since you are a master of sim building I have one that relates to this thread I would like to see.

Star =1.45 Msol
6 planets=0.5Mjup
Placement of planets such that

#1 @ 2.6 AU Base planet

#2 @ LG1 of Star and #1
#3 @ LG2 of Star and #1
#4 @ LG3 of Star and #1
#5 @ LG4 of Star and #1
#6 @ LG5 of Star and #1

All Should be in 1:1 resonance with #1
Just to see what happens.

 
Gravity Simulator does not provide an easy way to do this.  So this is a good simulation to practice your algegra and trig.  Start with a drawing of your system:

Now all you need to do is compute the x,y positions (call them r), and the x,y velocities (call them v).
 
Given is that m1 is 2.6 AU from the star.  m3, m4, and m5 are the same distance too:

 
The distance from the star to m1 and m2 can be computed by adding and subtracting the m0's Hill Sphere radius:

 
Now we need to break these distances up into their x and y components.  This is easy for 0, 1, 2, & 3 as they lie on the x-axis.  Just pay attention to the diagram to make sure you get your signs correct:

 
This requires a little bit of 4th grade trig for objects 4 & 5:

 
Now we need the velocities.  Objects 0, 3, 4, & 5 can be computed using the circular velocity formula:

 
Now we need to break these velocities up into their x and y components.  This is easy for 0 & 3 as they lie on the x-axis:

 
And requires a little bit of trig for 4 & 5:

 
Now we need the velocities of objects 1 & 2.  The circular velocity formula won't work here as object 1 will travel a little slower than normal since it is in the L1 point and must have the same period as object 0. And object 2 will travel a little faster than normal since it is in the L2 point and must have the same period as object 0. So we can get its velocity by dividing distance travelled by its period.  Its distance travelled is simply its distance from the star times 2 pi:

 
Now we need to break the velocities into their x and y components.  Since they lie on the x-axis, this is easy:

 
This gives us everything we need to start plugging in the numbers:
r1=388954463796.6-388954463796.6*(9.49304340314795E+26/(3*2.88392952200079E+30))
^(1/3)=370333651756.755
r2=388954463796.6+388954463796.6*(9.49304340314795E+26/(3*2.88392952200079E+30))
^(1/3)=407575275836.445
 
r4x=388954463796.6*cos(60)=194477231898.3
r4y=388954463796.6*sin(60)=336844446563.21
 
v0=v3=v4=v5=sqr((G*(9.49304340314795E+26+2.88392952200079E+30))/ 388954463796.6)=22245.5130780415
 
v4x=-v5x=22245.5130780415*sin(60)=19265.1794458029
 
v4y=v5y=22245.5130780415*cos(60)=11122.7565390208
 
v1=370333651756.755/sqr(388954463796.6^3/(G*(9.49304340314795E+26+2.883929522000
79E+30)))=21180.5310394943
v2=407575275836.445/sqr(388954463796.6^3/(G*(9.49304340314795E+26+2.883929522000
79E+30)))=23310.4951165886
 
 
Now make a table of each objects position and velocity vectors:
rx(0)=388954463796.6
ry(0)=0
vx(0)=0
vy(0)=22245.5130780415
 
rx(1)=370333651756.755
ry(1)=0
vx(1)=0
vy(1)=21180.5310394943
 
rx(2)=407575275836.445
ry(2)=0
vx(2)=0
vy(2)=23310.4951165886
 
rx(3)=-388954463796.6
ry(3)=0
vx(3)=0
vy(3)=-22245.5130780415
 
rx(4)=194477231898.3
ry(4)=336844446563.21
vx(4)=-19265.1794458029
vy(4)=11122.7565390208
 
rx(5)=194477231898.3
ry(5)=-336844446563.21
vx(5)=19265.1794458029
vy(5)=11122.7565390208
 
 
Open Gravity Simulator, create a system with a star 1.45 solar masses, place 6 objects, each 0.5 Jupiter masses in orbit around it, and use Objects > Edit Objects to enter the position and velocity vectors.
 
You will end up with
http://orbitsimulator.com/gravity/simulations/lagrange5.gsim
Before you run it, take your best guess as to what will happen.  I took my best guess, and I was correct  Wink
 
Gravity Simulator uses the programming convention that down is positive in the y-axis, while my diagram above used the standard math notation that up is positive in the y-axis, hence this diagram is inverted.

Back to top
 
 
Email View Profile WWW   IP Logged
abyssoft
YaBB Administrator
*****


I love YaBB 1G -
SP1!

Posts: 302
Re: Suppose that ....
Reply #7 - 05/15/07 at 10:16:33
 
WOW  Shocked
 
I definately did not expect that configuration to be that unstable.  It collapsed in less then a year.
 
The 4 planet configuration theat you mentioned earlier that might be stable collapses in less then 100 years.  undecided
 
The 3 planet configuration appears completely stable unless even the small perturber is present.  Ie eventually the trojan asteroids will eventual be booted.
 
The 6 planet configuration is also stable unless even the small perturber is present.  This is actually more sensitive to pertubations.
Back to top
 
 
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #8 - 05/15/07 at 11:14:00
 
Wow , that was quick !  
I started the sim Lagrange5 and saw as abyssoft mentions that the planets escape . Lowering the iteration time (up to 8 sec after first running at 64  ) gave somehow other pictures , so I decided to run it with Picard Integrator . The diameters were set to zero , so no collision takes place .  
Heres a picture of the start of the simulation after some 700 days .  
The system was further run for 80 years and shows a lot of chaos . It seems however that some planets are bound to each other , as the iteration step is decreasing periodically . It looks as if the m0 and m2 build some kind of an earth-moon system with high eccentricity ....
The interference of both can be seen where the dots became black .  
Has anyone yet come up with ...euh...a negative SMA ? I did here and I don't know what it means !!.  
Can it be a curvature away from the sun ?
Back to top
 

Lagrange5begin.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #9 - 05/15/07 at 11:26:29
 
[quote author=abyssoft link=1175890089/0#7 date=1179249393]WOW  Shocked
The 4 planet configuration theat you mentioned earlier that might be stable collapses in less then 100 years.  undecided
quote]

What timestep did you use in your sim and what accurancy of the initial velocities ? If the initial velocities are not accurate up to 1m/s or less a planet can quickly gain a lot of distance over 1 year .  
Back to top
 
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #10 - 05/15/07 at 12:42:44
 
I made some additional calculations about the 4 Earths at 90° . Data were generated with the Picard integrator ( same data as above ) . I wondered how the planets stay related to each other .  
Planet 1 is at 0° , Planet 2  is at 90° aso ... initially . From the picture hereunder it is clear that after some time (1000 years ) planet 2 moves away , planet 4 gets closer ,  
then they change : planet 2 gets closer , planet 4 goes away . But then ... (after 1250 years ) planet 2 comes closer , also planet 4. Then they get both further away ... The cycle seems to repeat somehow . I looks like planet 1 is the middle of an "accordeon" . Both planets approach and get further away ...aso  
Distances are expressed in AU .
The minimum distance is about 0.3 AU . This corresponds with ca. 17°
Remarkable is the cycle , some 330 years ...
Back to top
 

4Earths90degreesDistances.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #11 - 05/15/07 at 22:55:30
 
Concerning the Lagrange5 Sim : this system is chaotic . As mentionned I ran it with the P-Integrator , setting the diameters to zero , so no collision occurs .  
Herunder are some screenshots :  
Especially the motion of m0 and m2 are hard to describe : they obtain fabulous accelerations and stay close to each other , by while getting as close as 288 km !!; later  even closer . In the real world this isn't possible of course , but as they are point massas  they must accelerate then with :  
6.6E-11*9.5*E+26 /288000^2 is ca. 780km/s2 .  
this is fabulous . I think thats the reason why the system is hard to stabilize.  
In reallity the worlds are breaking down as they sweep around each other .  
The other half jupiters remain more or less in their orbits , although some are exchanging orbits ( see yellow and brown SMA line ).
Back to top
 

Lagrange5.gif
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #12 - 05/16/07 at 02:14:49
 
Quote from abyssoft on 05/15/07 at 10:16:33:
...The 3 planet configuration appears completely stable unless even the small perturber is present.  Ie eventually the trojan asteroids will eventual be booted...

The 60-60-60 configuration is quite stable, even in the presence of perturbers.  Saturn's moons, Tethys, Telesto, and Calypso are an example.  The 60-60 configuration is also stable.  
 
The 120-120-120 configuration (delete objects 0, 1, & 2) is stable, and will last indefinately unless a perturber is present, as is the 60-60-60-60-60-60 system.  
 
The [object 0, 1, & 2] system is stable unless a perturber is present, however, even the smallest pertabution, which an n-body simulation will induce, will de-stabalize it.  So objects in the L1 and L2 points usually won't last even a single orbit.
 
Moral:  in nature we won't find:
  • any objects in the L1 or L2 or L3 points of any other object.  
  • any triple system in the 120-120-120 configuration.  
  • any 6-body system in the 60-60-60-60-60-60 configuration  

But we may find pleanty of:
  • 60-60 systems
  • 60-60-60 systems

 
Quote from frankuitaalst on 05/15/07 at 22:55:30:
Concerning the Lagrange5 Sim : this system is chaotic...

Yes, this thing collapsed under chaos.  That's why I don't see any reason to follow the system after the collapse.  Give it slightly different starting conditions (run it at a different time step for a brief period of time), and you'll see drastically different results.   I imagine that sometimes you'll get orbiting pairs, sometimes you won't.  I would guess that even the orbiting pairs are short-lived over the course of 10s of thousands of years.
 
If you want to get an intuitive feel for what the L1 point is, consider the following:  The L1 point is stable in 2 axes, and unstable in 1 axis.  Picture 3 axes, one that is tangent to the orbit, one at a right angle to this, towards and away from the Sun, and one that is up/down (z-axis), hence you can't see it in the x-y plane diagrams in this thread.  This point is stable on the z-axis, and on the axis in the direction of motion, but unstable on the solar radial axis.
 
If perturbed on the solar radial axis, its drift will accelerate and it will quickly depart the L1 region.  But if perturbed on the axis tangent to the orbit, or on the z-axis, it will oscillate about this axis.  When it oscillates about both of these axes, it is said to have a Lissajous orbit.  The SOHO space telescope has a Lissajous orbit.
 
An object in the L1 point will feel an acceleration towards the Sun, and 180 degrees away,towards the Planet.  The star's pull is much stronger.  Adding these vectors together, it feels a pull towards the star that is slightly weaker that what it would normally feel for its distance, making the star seem "appearantly" less massive.  Objects in orbit around less massive stars move slower in circular orbits than objects in orbit around more massive stars.  When the "appearant" mass of the star matches what the circular velocity formula says it should be for an orbit with the velocity of Planet 0, you are in the L1 point.
 
This is as close to a video game as Gravity Simulator might come:

  • Pause the simulator
  • Open lagrange5.gsim
  • Delete Planets 2, 3, 4, 5
  • Edit Planet 1 and set it's mass to 0.
  • Open a thrust box: menu View > add Thrust Box
  • Open a distance and velocity box: menu View > add Distance and Velocity box
  • In both boxes, Choose Planet 1, with a reference object as the Sun
  • Unpause the simulation
  • Adjust the zoom so you can see Planet 0 and Planet 1
  • Increase the time step to 32 seconds.  As you get better, you'll be able to go even higher
  • On the thrust box, using ONLY the TOWARDS and AWAY buttons, attempt to keep the radial velocity as close to 0 as possible
  • If you are successful, Planet 1 will orbit the Sun with the same mean period as Planet 0.  If you drift ahead of, or behind Planet 0, you will naturally be pulled back due to the stability on this axis.  
  • See how many orbits you can complete.
  • When you get tired of doing this, simply stop correcting the orbit, and watch how quickly your object departs the L1 point.
  • When you're in the mood to do this again, try instead with an object in the L2 point.  Same instructions, except delete Planets 1,3,4, & 5, and use the trust box and distance box for planet 2.

 
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #13 - 05/16/07 at 13:52:42
 
Googling for some Lagrangian motions I found this link  
http://articles.adsabs.harvard.edu/full/seri/AJ.../0067//0000162.000.html
It's about  the "Klemperer Rosette" theory. Haven't read it yet fully but seems interesting as it is all about stability in this particular configuration . Seems that under certain conditions of massas a uniform n body configuration can be dynamically in an orbit around central body . The 90° Earths sim above is a special case .  Seems , as far as I've read it ,that a configuration in a regular polygon is dynamically stable ...
And here's an Java Applet about the same Klemperer Rosette :
http://www.burtleburtle.net/bob/physics/kempler.html
Back to top
 
« Last Edit: 05/16/07 at 17:38:51 by frankuitaalst »  
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #14 - 05/17/07 at 13:22:19
 
Heres the GravSim Simulation with 5 Earths ,72° separated from each other in the Klemberer configuration . The sim runs best at ca. 32000 s timestep . I used the rotating frame with Planet 1 as reference . It's nice to see how the system (other planets ) come close to planet 1 and then get away , decreasing the arc of 360° to 120° , then increasing again . Odd dynamic stability ...  
 
Modification : tried to upload the sim , but got a message this isn't possible , instead a screenshot of the system after ca. 1500 years  
Back to top
« Last Edit: 05/17/07 at 14:40:47 by frankuitaalst »  

5Earths72degrees.gif
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #15 - 05/17/07 at 16:29:32
 
Quote from frankuitaalst on 05/17/07 at 13:22:19:

Modification : tried to upload the sim , but got a message this isn't possible , instead a screenshot of the system after ca. 1500 years  

Try it again.  I just added .gsim to the allowable extensions.
 
I'm surprised this system is stable.  I'm anxious to try it.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #16 - 05/17/07 at 23:51:34
 
It works , thanks . Sim is running now for 25000 years ...very odd dynamics...but you have to wait for ca.600 years before the fun starts . It looks as if the planets are tied to each other with springs ..pulling and extending ...I haven't seen them com closer than 90° yet .
Back to top
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #17 - 05/18/07 at 00:51:43
 
By mistake I uploaded the 4 Earths at 90° instead of 5 Earths at 72° in the Klemperer configuration .  
Here is the 5 Earths sim . But both have the same type of dynamics ...
Back to top
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #18 - 05/18/07 at 01:06:37
 
Here' s a picture seen from above in the xy frame , centered on one Earth of the 4 Earths sim after 6000 years. All other planets come close to each other and go away aso. The more black the more a planet spends in this region . The innermost circle represents the orbit of the sun (1AU) .
Back to top
 

Klemperer4Earths90.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #19 - 05/18/07 at 02:44:35
 
Running the Klemperer System with 4 Earths for a longer time (80000 y ) gives the following evolution of the SMA's :  
The system seems to behave odd , a lot of variation in SMA , but they al stay close within 4/Thousand.  
Plotting the mean SMA gives almost a straight line .  
Is there some correlation between mean SMA and the energy of the system .  
Zooming in into the 2 opposite planets in the 1000-3000 years gives the plot underneath .  
The planets evolve in a "regular" but unpredictable way . I honestly don't understand the behaviour , but I agree with Tony that Earth must be protected somehow against collisions by the horseshoe mechanism .
Back to top
 

Klemperer_4_Earths_90.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #20 - 05/18/07 at 12:34:12
 
Is the 4 Klemperer system stable under foreign influence ? To find out Jupiter was added to the simulation . After 6000 years of simulation the system seems to behave as before . The mean SMA however varies more than before ( not visible in the graph , but visible if the SMA Mean is put on a second axis ) .
Back to top
 

Klemperer_4_EarthsJupiter.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #21 - 05/19/07 at 00:29:43
 
Is the Klemperer system stable when the planets have different mass ?  
under certain conditions it is .  
Hereunder is a link to the 4 Planets sim .  
1 Planet was given a mass of 3 Earths , the other 0.33 Earths , so quite a difference . As a seed the planet 1 was positionned 5 km "north" .  
The plot heraunder gives the evolution of the SMA of each planet . Interesting is the second plot , zooming into the first years . The system seems to  react immediatelly . The planet in front slows down , the planet behind acclerates , the come close to the big planet ,they are both repelled , ga away , come closer again ... The symmetry is lost after 200 years .  
 
The GravSim (second post herunder ) is even funnier when run in a rotating frame on planet 1. I ran it at 8000-32000 seconds . Watch the planets 2 and 4 coming close and moving away ...
But , wait , take a look then at planet 3 ( opposite of the big ) . It stays a long time at home position , then moves at big speed towards planet 2.....
Negative gravity ?  Shocked  
Back to top
 

4Klemperer_one_big.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #22 - 05/19/07 at 00:31:08
 
Heres the GSim file .
Back to top
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #23 - 05/19/07 at 10:52:47
 
May it be possible for Earth to host Klemperer Moons ?  
Although planets can theoretically share other planets in their orbit , it is difficult for Klemperer Moons in an orbit around Earth to survive .  
Starting with 4 Moons , each with a mass of 1/10 of our moon and gradually decreasing the mass in different simulations the result is  instability resulting in eccentric orbits and collisions .  
As the mass of the moons decreases the lifetime seems to go up .  
The system with 4 moons , each 1/1000 Moon mass survived for 37 years . Other heavier systems crashed after 3 years . The picture shows the orbits of the moons ,each 1/1000 Moon massas , centered on earth .  
Back to top
 

KlempererMoons41o1000.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #24 - 05/30/07 at 13:33:09
 
Simulating the 5 Earth Klemperer configuration in a rotating frame gives the following result .
All the planets share the same orbit around the sun , but are 72° separated from each other .  
The first earth is initially situated at 3 o clock . This is the reference planet in the rotating frame.  
 
After a couple of hondred years the planets come into motion relative to each other and move towards the first earth .  
They increase their speed , come close and ....reverse.
This goes on and on .  
The individual planets are not easy to distinguish as they share the same orbit , except for their color .  
The sim spans more than 3000 years ...
Back to top
 

5Earths72degreesRot.gif
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #25 - 05/30/07 at 18:00:50
 
I'm not sure what's going on here.  When I try similar simulations, they stay stable for a few hundred years, or more, depending on the time step.  Theoretically, they should be stable indefinately.  So errors introduced by the integration are what cause it break down with time.  But when it breaks down, it should become very chaotic.  They should not stay in the same orbit.  They should become eccentric and they should make close approaches to each other.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #26 - 05/31/07 at 09:38:36
 
Perhaps the most simple simulation is this one : 1 Sun and 2 Earths .  
Both Earths share the same orbit but the second is exactly 180° ahead of the first . The system seems stable and is performing as the 5 Earth system .  
In order to investigate the stability I added Jupiter to the sim .
Will it still be stable ?  
Here's the result in a rotating frame around Earth1 . The system seems remarkable stable for the first 3000 years .  
There is no indication for instability .
Back to top
 

2Earths180JRot.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #27 - 05/31/07 at 11:24:14
 
The system above seems to have a remarkable dynamic stability . In the run for ca. 6000 years there seems to be a periodicity of ca.700 years , where the planets approach each other .  
Closest approach is at ca. 24°.  
The second planet spends a lot of time at 180° and then slowly moves away , gaining speed until it gets as close as 24° and then moves the other way back .  
Once arrived at 180° it can ...continue after a while or... return .  
So it sometimes "attacks" from the other site .  
Have no idea were the cycle period nor the 24° come from and how they are related to the system parameters.
Back to top
 

2EarthsJupGraph.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #28 - 05/31/07 at 14:30:50
 
The symmetry in the above system is remarkable as can be seen from the plot hereunder .  
Starting at an angle of 180° one tends to slow down , the other is gaining speed . As a result he angle betweem them becomes smaller .  
 At a point at about 50° they are slowing down and speeding up and come as close as 24 degrees , then are moving away again ...
Both planets seem symmetric to each other
Back to top
 

2EarthsJupSMA.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #29 - 08/11/07 at 03:15:10
 
What is happening in this simulation ?  Huh
The sim herunder consists out of two solar masses ( Sun and the outer orbit ) .  
They orbit their common barycenter . Distance between both is 1.5 AU.  
So two suns which are on the Earth orbit .  
A really small excentricity is added in their orbits , just as Earths orbit around the sun . There is no central sun .  
A third body is added then with mass 0.03 solar masses . The planet shows itself as the inner circle and is placed just in between the 2 solar masses , also in their barycenter .  
The screenshot is centered on the first sun ( indicated by Sun) .  
One would think this creates a stable orbit for both suns , but see what happens .... Shocked
The orbits of the suns change quite a lot ...
They seem to be stable for long , and then suddenly they "jump"...This is repeated several times.  
At the end of the simulation the planet is ejected .  
Fysically impossible ? Resonance ? Well yes , a kind of .  
I'll post the .gsim file . This may explain a lot .
Back to top
 

2Sunsandaplanet.gif
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #30 - 08/11/07 at 17:06:53
 
Quote from frankuitaalst on 08/11/07 at 03:15:10:
I'll post the .gsim file . This may explain a lot .

If you do that, I'll take a look.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #31 - 08/11/07 at 17:54:02
 
Sorry , but for some reason I get the "6 overflow" message . I use the Beta22april2007 version .
Back to top
 
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #32 - 08/12/07 at 03:00:00
 
Heres the clue to the mystery above :  
The sim herunder shows this system edge on .  
What I didn't tell was that the planet was positioned originally high above the ecliptic of the sun orbits , right in the middle  embarrassed
One can see the suns react on a still unvisible planet . As the planet crosses the plane the suns "jump".  
The planet gets an overshoot , returns ...and so on .  
Seems to act as a "cosmic spring".  
I didn't succeed in putting this system in GravSim as the system is referenced in global coordinates.
Back to top
 

2SunsplusplanetEdge.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #33 - 08/18/07 at 07:55:40
 
On the stability of a Lagrange orbit ...
Suppose our Earth had a sister-planet in its Lagrange point L4 or L5 .
What happens ?  
Not much , but ...
In the frames herunder the movement of both Earths is shown in a rotating frame of our Earth .  
Earth is at 3 'o clock , the sister planet is at 1 'o clock .  
The simulation starts with 1 Earth , showing also Mercury and Venus .  
The next frame shows the same simulation but with 2 Earths .  
Then the massas of the Earths are increased by 10, 100 ,1000 , 2000 , 4000, 8000...
Each picture shows the system after 250 years.  
The system gets unstable between as the sum of both masses  is somewhere in between 1.2% and 2.4% of the solar Mass.  
Running the application it seems that Venus plays a big role in the unstability of the system .  
At a given moment she really can't decide which Earth to visit first . Smiley
The last picture shows the last frame for above where Mercury , Venus and 1 Earth are ejected .  
Remarkable is the amazing orbit of Mercury in the 2*4000 Earth Mass simulation.
Back to top
 

LagrangeEarthMass.gif
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #34 - 08/18/07 at 12:27:57
 
It's about 4%, the mass ratio between the Sun and the combined masses of the double Earths.  Here's a link from the BAUT forum where we discussed this:
http://www.bautforum.com/astronomy/58590-orbit-earth-changes.html
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #35 - 08/19/07 at 12:07:12
 
Nice work in the above link in the BAUT-forum !  
The abstract of Chambers and Laughlin stipulates for the linear stability : 2mpl/(2mpl+Msun)< 0.03812 .  
http://adsabs.harvard.edu/cgi-bin/np...2L&db_key=AST&
 
This number of about 4% (0.03812) is AWFULLY correct as can be seen in the frames hereunder .  
4 sims were run with increasing mass of Earth so that the ratio becomes : 0.03792; 0.03811 ; 0.03820 and 0.03825 ( compare to 0.03812 ) for more than 1000 years .  
The first two are stable ( ratio = ok ) , Earths hardly move relatively .  
The third run is just above the ratio , here there is a dynamically stability , not a linear one .  
In the last run the ratio is exceeded by 0.00013 and the stabiltity is completely gone .
Back to top
 

Lagrange2Earths3.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #36 - 08/20/07 at 11:30:49
 
Lagrangian points :  
Is a system which has a sister planet in its Lagrangian point L4 or L5 ( also at 60° behind or ahead ) capable to have also other bodies moving in an horseshoe orbit ?  
IT IS !  
Consider an Earth-Sun system , add a second Earth at 60° and add two minor bodies , starting at the opposite site of the sun , vieuwed from Earth . Give them a position of resp. 1.01 Au and 0.99 AU .  
So they start close to each other , but one is nearing the Earth , the other goes away from Earth .  
Here' s what happening in a rotating frame .....
 
Both planets orbit in a horseshoe , but whats even more : they interchange orbits after each close approach .  
The both Earths are the little dots at 1 and 3 'o clock .  
The system is sensitive to the starting conditions : bodies with more initial distance (> 1.02 AU /0.98 AU fi) tend to behave in a normal orbit .  
 
One can imagine what other civilizations living in such a system would see : a big body , as bright as Venus always staying at the same  distance from the sun , and then one smaller body which approaches and then goes away again ; then another similar  body approaches , which goes away again ...
Back to top
 

2EarthsLagrangeplushorseshoe.gif
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #37 - 08/21/07 at 10:13:00
 
Here' the GravSim .gsim file of the above system .  
The system should be unpaused and the view set to rotating frame around Earth1.  
For some reason I can't get the rotating frame synchronised with Earth1. Can Tony take a look at it ?  
Back to top
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #38 - 08/21/07 at 11:02:17
 
Your 2 Earths do not start out 60 degrees away from each other.  They're spaced only about 52 degrees, which is not enough for a Lagrangian lock.  So when they pass each other, they affect each other's semi-major axis, and hence, each other's periods.  That's why you can't get a lock on the rotating frame period.
Back to top
 
 
Email View Profile WWW   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #39 - 08/21/07 at 11:18:08
 
Try this:  http://orbitsimulator.com/gravity/simulations/trojanEarths.gsim
You get 2 Earths spaced 60 degrees, 2 trojan asteroids, one orbiting L4, and the other L5, a horseshoe asteroid, an an asteroid caught between the 2 Earths.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #40 - 08/21/07 at 11:22:23
 
Thanks , I changed the initial conditions , put again the Earth2 at 60° and ran the application .  
I can now stabilise the Earth1 , but the Earth2 moves away ...(???) . It seems to form a 9 corner polygon .
I don't understand .  
Here"s the sim :
Back to top
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #41 - 08/21/07 at 11:27:11
 
Quote from Tony on 08/21/07 at 11:18:08:
Try this:  orbitsimulator.com/gravity/simulations/trojanEarths.gsim
You get 2 Earths spaced 60 degrees, 2 trojan asteroids, one orbiting L4, and the other L5, a horseshoe asteroid, an an asteroid caught between the 2 Earths.

This works ! was it an existing simulation ?  
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #42 - 08/21/07 at 11:59:47
 
No, I just made this now:
 
File> New
Objects > Edit Object, rename Object 1 to Sun, change color to yellow
 
Objects> Create Objects
Mass: 1 Earth mass
SMA: 1 AU
Change all the 100% in the +- boxes to 0%
Mean Anomoly: 180
Choose Color
 
Repeat, except Mean Anomoly: 120
 
Objects> Create Objects
Number of objects: 50
SMA: 1 AU
Mean Anomoly: 180 +- 100%
 
Save the sim.
 
You now have 50 asteroids sharing an orbit with 2 Earths.  Choose one asteroid from each catagory and delete the rest.
Back to top
 
 
Email View Profile WWW   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #43 - 08/21/07 at 12:07:24
 
Quote from frankuitaalst on 08/21/07 at 11:22:23:
Thanks , I changed the initial conditions , put again the Earth2 at 60° and ran the application .
I can now stabilise the Earth1 , but the Earth2 moves away ...(???) . It seems to form a 9 corner polygon .
I don't understand .
Here"s the sim :  

In this sim, Earth 2 doesn't have the same semi-major axis as Earth 1.  It is 10 million km closer to the Sun, so they'll never lock into a 1:1 resonance.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #44 - 08/21/07 at 12:20:30
 
Some input error I think , inputting the X,Y coordinates . Your approach with the automatic generation is a lot quicker and gives accurate results . Smiley
Back to top
 
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #45 - 08/25/07 at 14:59:42
 
Concerning the two horseshoe objects above in the reply nr 36 : they interchange orbits and seem to be in close counterphase to each other . ( when one is in the outer orbit the other is in the inner )  
The last statement is not true as can be seen from the plot hereunder , where the distance to the sun and the velocities are represented .  
Both bodies have a cycle of about 115 years , but there seems to be a phase lag of about 15 years between the time the bodies change the orbit . This means there are times where they are at the some distance to the sun ( but at this moments they are about 360-~60 =~300 degrees separated ) .
Back to top
 

fsscr068.jpg
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #46 - 08/27/07 at 11:28:27
 
The orbital exchange happens when they are near each other, and can alter each other's orbits.  This graph is similar to the graph of Janus and Epimetheus, except Janus and Epimetheus do not have similar masses:
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Suppose that ....
Reply #47 - 08/27/07 at 12:19:18
 
Is there a third body involved in the case of Janus and Epimetheus ?  
In the case of the two asteroids in the Lagrangian orbit I have the impression that the change takes place when one body is near to one Lagrangian body (look at the rotating frame in the beginning ) ; if the 2nd ast approaches the other Lagrangian body this asteroid also changes orbit .
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1051
Gender: male
Re: Suppose that ....
Reply #48 - 08/27/07 at 12:25:18
 
Sorry, I forgot you were referring to 3 bodies sharing an orbit.  Janus and Epimetheus are only 2.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Twin Earth System
Reply #49 - 05/11/08 at 12:29:42
 
I've been playing aroud with a twin Earth-Moon system , placed right opposite to our sun , so at 180° .  
Adding a small exces velocity  of 1m/s makes the system move after about 100 years . The result may be seen here in a rotating frame aroud Earth. The second moon was made invisible .  
The system gets in a horseshoe orbit to Earth . It has tendency to approach Earth from left to right alternatively .
Back to top
 

SecondEarth.gif
Email View Profile   IP Logged
Pages: 1 
Send Topic Print