Welcome, Guest. Please Login.
Gravity Simulator
11/21/17 at 02:16:51
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
Procyon system and stability (Read 4978 times)
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Procyon system and stability
10/27/10 at 13:56:47
 
In the Baut forum there's a topic about the stability of an eventual planet around Procyon A .  
Here's a setup of the system Procyon with Ma= 1.42  Ms, M B= 0.6 Msun, ecc = 0.4 and period 40.3 years , resulting in a semi major axis of 14.85 AU .  
The 10 dummy planets in blue were positioned at  2.7 +/- 10% around Procyon A .  
Running the sim seems to give a stable system for the first 1500 years .
 
Hereunder the sim file
Back to top
 
Email View Profile   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Procyon system and stability
Reply #1 - 10/29/10 at 12:20:55
 
Here's a simulation over more than 500 years of the Procyon system .  
Added are 10 bodies in a range of 3 AU +/- 5% . Most bodies seem stable , except the last one which eccentricity probably will rise further until it escapes . So the stability region may be around 3 AU for ths system .  
 
While running other simulations I noticed that once a planet escaped , then was captured by Procyon B , then escaped again to Procyon A, until it finally escaped in empty space .
Back to top
 

Procyon_Simulation.gif
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Procyon system and stability
Reply #2 - 10/30/10 at 01:05:47
 
I've got a spreadsheet that calculates the stable orbits for binary systems, as per this paper:  
Long-Term Stability of Planets in Binary Systems (Holman, Matthew J.; Wiegert, Paul A.) The Astronomical Journal, Volume 117, Issue 1, pp. 621-628.
 
Entering the parameters that you stated, the maximum stable orbit around Star A should be 2.72 AU, and the maximum stable orbit around star B should be 1.65 AU.  
Planets can orbit the centre of mass of both stars at a distance of 53.72 AU or beyond.  
 
Looks like your simulation is confirming that! (I wonder if the planet whose orbit is getting more eccentric in the simulation started off beyond 2.7 AU?)
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Procyon system and stability
Reply #3 - 10/30/10 at 05:22:38
 
Thanks for the link Mal .  
The Formula looks like this :
 
ac = [(0.464 ± 0.006) + (−0.380 ± 0.010)μ + (−0.631 ± 0.034)e +(0.586 ± 0.061)μe + (0.150 ± 0.041)e2 + (−0.198 ± 0.074)μe2]ab
 
If I plug in the values of the Procyon system I get indeed 2.71AU as the nominal .  
However the minimal and maximal values for AU seem to be : 2.12 AU and 3.31 AU  
The simulation tends to indicate that the outermost body at 3.15 AU is instable , which  corresponds with the formula .
Back to top
 
 
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Procyon system and stability
Reply #4 - 10/30/10 at 12:05:42
 
Quote from frankuitaalst on 10/30/10 at 05:22:38:
Thanks for the link Mal .
The Formula looks like this :

ac = [(0.464 ± 0.006) + (−0.380 ± 0.010)μ + (−0.631 ± 0.034)e +(0.586 ± 0.061)μe + (0.150 ± 0.041)e2 + (−0.198 ± 0.074)μe2]ab

If I plug in the values of the Procyon system I get indeed 2.71AU as the nominal .
However the minimal and maximal values for AU seem to be : 2.12 AU and 3.31 AU
The simulation tends to indicate that the outermost body at 3.15 AU is instable , which  corresponds with the formula .

 
That's the equation I use...  
 
I don't know if I've got μ right - the paper says that is mu1=m2/(m1+m2). When I'm calculating the critical distance for Star 1 I put that value for mu in the equation. When I'm calculating the critical distance for Star 2, I use mu2=m1/(m1+m2) instead. Is that correct? I doesn't make sense to me to use the same value for mu if it's calculated like that, but now you've got me wondering.  
 
I've seen it expressed elsewhere as (M1*M2)/(M1+M2) as well. That'd give me a constant mass ratio to use, but it's different to mu1 and mu2 (in this case it'd be 0.422, which is about the same as M2/M1).  
 
Sigh. Why do they have to make something as simple as a mass ratio so complicated?!
Back to top
 
 

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


I love YaBB 1G -
SP1!

Posts: 302
Re: Procyon system and stability
Reply #5 - 10/30/10 at 12:48:23
 
I wonder if the formula holds for retrograde orbits?
Back to top
 
 
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Procyon system and stability
Reply #6 - 10/30/10 at 14:08:20
 
Quote from Mal on 10/30/10 at 12:05:42:
right - the paper says that is mu1=m2/(m1+m2). When I'm calculating the critical distance for Star 1 I put that value for mu in the equation. When I'm calculating the critical distance for Star 2, I use mu2=m1/(m1+m2) instead. Is that correct? I doesn't make sense to me to use the same value for mu if it's calculated like that, but now you've got me wondering.

I've seen it expressed elsewhere as (M1*M2)/(M1+M2) as well. That'd give me a constant mass ratio to use, but it's different to mu1 and mu2 (in this case it'd be 0.422, which is about the same as M2/M1).

Sigh. Why do they have to make something as simple as a mass ratio so complicated?!

 
You use the formula correct Mal . The paper says the perturber is the other body . So , if orbiting around M1 , the perturber is M2 , and mu= M2/(M1+M2) . If orbiting around M2 then mu = M1/(M1+M2) .  
To Abysoft : the formula definitively is only valid for prograde orbits .
Back to top
 
 
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Procyon system and stability
Reply #7 - 10/30/10 at 14:15:18
 
OK, phew! So I was doing it how it's supposed to be done.
 
So I wonder why your numbers are different to mine? Or were you talking about the 'minimal and maximal orbits' in your simulation, rather than calculated by the paper?
 
I'm not sure if any planets would be retrograde anyway, I thought you needed some serious Kozai mechanism going on to flip the orbit like that?
Back to top
 
 

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


I love YaBB 1G -
SP1!

Posts: 302
Re: Procyon system and stability
Reply #8 - 10/30/10 at 14:46:10
 
http://news.softpedia.com/news/Binary-Star-System-Can-Support-Retrograde-Planets -159688.shtml
 
Outlines that there are at least 2 such retrograde planets and that they are more likely to survive in a binary system then prograde planets.
Back to top
 
 
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Procyon system and stability
Reply #9 - 10/30/10 at 16:09:18
 
Quote from Mal on 10/30/10 at 14:15:18:
OK, phew! So I was doing it how it's supposed to be done.

So I wonder why your numbers are different to mine? Or were you talking about the 'minimal and maximal orbits' in your simulation, rather than calculated by the paper?

I'm not sure if any planets would be retrograde anyway, I thought you needed some serious Kozai mechanism going on to flip the orbit like that?

Your numbers are the same as mine Mal . 2.72 AU .  
I only interpreted the +/- signs of the equation to estimate the lower and upper bound .  
About the Kozai : yes , but there's another possibility maybe .  
As I mentionned above I saw several planets change their orbits from M1 to M2 several times . It might be possible that when the planet returns to the other sun I may choose the other direction , becoming a retrograde planet ...
I say maybe because I didn't pay attention well enough when I observed the transition  
Back to top
 
 
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Procyon system and stability
Reply #10 - 10/30/10 at 18:23:06
 
Quote from frankuitaalst on 10/30/10 at 16:09:18:
I only interpreted the +/- signs of the equation to estimate the lower and upper bound .

 
Oh, right - I just ignore those when doing the calculations (technically you're correct though, it's just easier to go with the mean value Wink ).  
 
Quote:
As I mentionned above I saw several planets change their orbits from M1 to M2 several times . It might be possible that when the planet returns to the other sun I may choose the other direction , becoming a retrograde planet ...

 
How likely would that be though? It's hard enough to capture a planet in the first place isn't it?  
Is this system likely to have a strong enough Kozai mechanism to pump up the planet's inclination enough to flip it?  
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Procyon system and stability
Reply #11 - 10/31/10 at 16:04:06
 
The reversal of motion is very rare I think .  
Hereunder is a simulation with initial conditions at 3.3 AU and inclination of 10° +/- 5% ( 20 bodies ) .  
It's interesting to see how eccentricities are pumped up . Some planets wander to the B star and eventually return .  
I've marked the body 17 in red which will get in a retrogarde orbit after about 500 years .  
 
Edit : my mistake : I saw I've posted thye wrong sim . I was playing with several sims . The red planet in the above sim does not evolve to retrograde .  
Unfortunatelly I forgot to save the right one  embarrassed
Back to top
« Last Edit: 11/01/10 at 10:27:47 by frankuitaalst »  
Email View Profile   IP Logged
Pages: 1
Send Topic Print