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Stability of orbits in case of inner planets (Read 5019 times)
Belegaer
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Stability of orbits in case of inner planets
09/14/06 at 03:46:32
 
Hello,
 
I've been experimenting with stability of planetary orbits in case of Sun-like stars. I was inspired to do so after reading few articles on habitability zones etc. I have been trying to establish how many Earth-like planets could possibly coexist on stable orbits within estimated habitability zone in case of stars like Epsilon Eridani - Sun - Procyon A.
 
To my suprise I got results suggesting that Venus and Earth could coexist as close as 15.000.000 kilometers in their mutual closest approach. I have run my experiment with zero eccentricity for both planets at the start time for about 2.000 - 3.000 years of simulation .
 
Could someone confirm/refute that this is realistic scenario?
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Re: Stability of orbits in case of inner planets
Reply #1 - 09/14/06 at 04:15:42
 
In order to truly determine stability one would need to run a simulation for about 10 million years at no more then 1024 per step.
 
But, you are quite right in that these 2 bodies can be rather close in astrometric terms and still have stable orbits, this is the result of gravitational binding pressure.  As long as the hill spheres do not overlap and are both bodies are at least 2 hlarger body in separation the system has a greater chance of stability.
 
Feasibly there could have been 8+ earth like dwarf planets around Sol in the HZ, vs. only 1 planet in the HZ and 2 planets on the periphery we actually have.
 
Gravitation binding pressure is an area being explored in M-Theory to better understand the nature of gravity.
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Re: Stability of orbits in case of inner planets10
Reply #2 - 09/14/06 at 09:08:00
 
In order to truly determine stability one would need to run a simulation for about 10 million years at no more then 1024 per step.  
 
I understand. However 10 million years is too much  Cheesy. I had to satisfy myself only with few thousand of years to get a clue about the properties of my version Sun/Earth/Venus system.
 
Sadly I must admit that I have heard about Hill sphere(s) for the first time (in the meantime I have checked wikipedia). Nevertheless, it was clear to me that the inner Solar sytem area is dominated by Sun damping the mutual perturbations of both planets whilst in outer parts of solar system the difference of 15.000.000 could be too small to keep interference of both planets sufficiently small.
 
It will be quite interesting to see the results of TPF/Darwin missions - how common are rocky planets in Sun-like systems.
 
Edited for clarity Belegaer
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Re: Stability of orbits in case of inner planets
Reply #3 - 09/17/06 at 08:24:13
 
Hallo Tony ,  
 
Running Cruithne I observed that the Mercury orbit perihelium rotated in a continious way .  
I wonder if this is due to a too big timestep (1024 or maybe 2048 ...) or are some relativistic effects incorporated in the programm ?  
 
Kind Regrards  
Frank Kestens
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Re: Stability of orbits in case of inner planets
Reply #4 - 09/17/06 at 08:35:07
 
The Procession of Perihelion as well Aphelion are a normal occurance in any multi bodied system where a primary is also in motion.  This will occur regardless whether or not relativism is taken into account or not.  The only thing that would change and not really noticable would be the frequency rate of the procession.
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Re: Stability of orbits in case of inner planets
Reply #5 - 09/17/06 at 12:28:13
 
The Cruithne simulation is in a rotating frame.  What you see is Mercury's perihelion precessing with respect to Earth's position rather than Mercury's perihelion with respect to Sun.  Try turning off the rotating frame. Click menu View > Rotating Frame to make the check mark go away.  Now you will notice that Mercury's precession is unnoticable unless you run the simulation for a long time.  In this case, you will notice precession in Mercury's orbit.  How much depends on your time step.
 
Newtonian physics predict that Mercury's orbit should precess 531 arcseconds per century.  But observations show that it actually precesses 574 arcseconds per century.  The extra 43 arcseconds are attributed to relativity.  Gravity Simulator does not take relativity into account.  It is purely Newtonian point mass.  Here's a graph showing Gravity Simulator's accuracy at different time steps for the precession of Mercury.  All simulations are 10,000 years.
 

 
At a time step of 2048, Mercury's perihelion only advances about 83 arcseconds per century.  But a vast improvement is seen by cutting the time step in half.  At a time step of 1024, the advance of its perihelion jumps to 414 arcseconds per century.  Continuing this trend, it approaches the correct answer of 531 with smaller time steps.  At time step 128, it is at 523 arcseconds per century.  It takes over a day to run a 10,000 year simulation at time steps slower than 128, which is why I stopped there.
 
At time steps greater than 2048, Mercury's orbit precesses backwards.
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