Ultimate space simulation software

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evildrganymede
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Solar System dynamics discussion thread

11 Aug 2019 08:05

These are really great articles, thanks! Question for you - You use "particles" to test orbits in the L4 and L5 points, do they have any mass? If not have we got any way to put anything with mass there to see what happens? There's a discussion about having planets in 'trojan orbits' (at the L4/L5) over on the Elite Dangerous forums and I'm wondering whether the mass of the "particle" would affect the stability of the L4/L5 orbits.
 
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Watsisname
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Solar System dynamics discussion thread

11 Aug 2019 16:31

evildrganymede, good question!  In all my calculations and figures I treat the particle as massless (a good approximation if it is say, a spacecraft or asteroid as a Trojan of a planetary body).  

If it is massless, then stability about the L4 and L5 points occurs if the primary mass is at least 24.96 times the mass of the secondary.  But what if the particle is not massless?  In generality, the condition for stability of orbits about L4 and L5 is given by

Image


where you may call m1 the star's mass, m2 the mass of the big planet, and m3 the smaller (Trojan) body.


With this formula we can quickly confirm that if we set m3=0, then m1 must be at least 24.96 times m2.  But what if we let m1 be 1 solar mass and m2 be the mass of Jupiter?  How big can m3 be and still be stable as a Trojan body?  


Answer:  About 40 Jupiter masses!  (Surprising, yes!)  

Even if both planets had the mass of Jupiter, then they could be stable in one another's orbital space, as Trojans of one another.  But they could not both have a mass of over 20 Jupiters, or else the stability would fail since together they'd be too massive relative to the star, and the chaos of the 3-body problem would appear.


In other words, for almost any reasonable choice of masses of star and planet, another planet could exist in a stable Trojan orbit.


Why then do we not find many Trojan planets in nature?

The analysis here has been for an idealized situation, where we considered only two masses (plus a co-orbital body) and for them to be in circular orbits.  Eccentric orbits will reduce the stability, as will the presence of additional planets in the system.  Many of the Trojan asteroids of Jupiter are not in stable orbits for example, and those asteroid groups have slowly been eroded over time.  Others may be newly caught there, and remain only temporarily.


But maybe an even bigger reason is that planetary systems are very dynamic, especially in their youth.  The planets form by accreting dust and gas out of the disk, and then their orbits migrate as they scatter nearby planetessimals away.  The planets also influence one another, potentially even leading them to swap places or get ejected from the system.  So even if a Trojan planet were to exist in a system's early history, it probably wouldn't remain there for very long while all this chaos is happening.
 
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evildrganymede
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Solar System dynamics discussion thread

11 Aug 2019 17:47

Thanks - so 40 MJ... that's actually very close to 1/24.96 of a solar mass (that would be 41.96 Jupiter masses exactly)? I'm assuming that's not a coincidence? So are you saying that generally speaking *the total mass of everything in the orbit* (the main planet and any trojans) has to be at least 1/24.96 of the star's mass for the configuration to be stable? So Jupiter could say have an earth-mass planet in its L4 and L5 points and that would theoretically work? Heck, it could have a saturn-mass planet in both those points and that'd still be stable? 
 
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Watsisname
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Solar System dynamics discussion thread

12 Aug 2019 01:02

evildrganymede wrote:
Source of the post 40 MJ... that's actually very close to 1/24.96 of a solar mass (that would be 41.96 Jupiter masses exactly)? I'm assuming that's not a coincidence?

Good observation!  It is very close and not by coincidence.  It's not quite correct that the rule is "total mass of everything in the orbit must be less than 1/24.96 of the star, but it's a decent rule of thumb for a wide range of masses.  For example if we set m2 = m3, and m1 = 1, then m1/(m2+m3) must be 25.23.  Slightly more than 24.96.  However, this is splitting very fine hairs, as anything close to these boundary conditions for stability will only barely be stable anyway.

evildrganymede wrote:
Source of the post So Jupiter could say have an earth-mass planet in its L4 and L5 points and that would theoretically work?

This I do not know.  It would be stable for Jupiter and an Earth in L4 or L5, but I'm not sure about an Earth in both simultaneously.  That would turn this from the unrestricted 3-body problem to an unrestricted 4-body problem, for which I haven't seen the criteria for stability of equilibrium points.  I'm sure there must be a paper about this somewhere, though.
 
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Watsisname
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Solar System dynamics discussion thread

20 Sep 2019 13:54

  • Common misconceptions about tides. What are tides really?
  • "The devil is in the detail" of tidal theory (coastlines, resonances, delays, tide-predicting machines, tides caused by planets, etc...).



This subject came up in the Astronomy Q&A thread, and I've made an attempt to explain the workings of the tidal interaction and its effects on both Earth and Moon there.  As the above implies, the details are actually incredibly complex, and I only give the very basic "schematic" idea.  Hopefully this is a good starting point.  There is much, much more that can be said about it!
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