Maximum number of habitable planets per system – article

[vec]

Thought you guys might enjoy this:

http://planetplanet.net/2014/05/23/buil ... -together/

[Particles the Greek]

Niiiiiiiiccccccceeeee!!! Well found.

[the duke of nuke]

Mm, I like this - nice find. Interesting comment he makes about it being a good setting for stories

[Ketumak]

Thanks for posting. I wouldn't want to do anything so planet-rich myself, I prefer to keep things more likely-seeming, but it's interesting to see just how far you can push things and still have a viable solar system.

[zompist]

He's designed the Firefly system.

(Whedon's Verse was designed to have a load of planets requiring no interstellar travel.)

[su_liam]

It would be interesting to try a few variations of this in Mercury6.

I'm somewhat dubious about the stability of these, particularly the trojan relationships. Even L-4 and L-5 orbits are only stable in the sense that LEO is stable. It still requires some impulse to maintain those orbits over time, just less than would be required for L-2 and L-3 orbits.

I suspect that the Jovian Trojan asteroid clusters are in some sort of dynamic equilibrium. Asteroids are captured into the cluster at about the same rate that they are lost, rather than retained in the Trojan positions for billions of years.

I'm trying to find free full-text of some of his supporting documentation. $35.00 an article is... a bit steep.

I wonder how much stability would vary with mass of the primary?

[Thomas Winwood]

I'm somewhat dubious about the stability of these, particularly the trojan relationships. Even L-4 and L-5 orbits are only stable in the sense that LEO is stable. It still requires some impulse to maintain those orbits over time, just less than would be required for L-2 and L-3 orbits.

I suspect that the Jovian Trojan asteroid clusters are in some sort of dynamic equilibrium. Asteroids are captured into the cluster at about the same rate that they are lost, rather than retained in the Trojan positions for billions of years.

I can't find a primary source (the best I have is this IAU circular) but there's a quote attributed to K. A. Innanen that "contrary to intuition, there is clear empirical evidence for the stability of motion around the L4 and L5 points of all the terrestrial planets over a timeframe of several million years". In addition, Wikipedia claims Mars trojan 5281 Eureka "is located deep within a stable Lagrangian zone of Mars, which is considered indicative of a primordial origin—meaning the asteroid has most likely been in this orbit for much of the history of the Solar System".

[gach]

It's a long time since I took any celestial mechanics but L4 and L5 should be stable against small perturbations. It might just be a bit misleading to think of them as strictly point like attractors. A small body trapped in either L4 or L5 can actually move quite far along its orbit from the nominal Lagrangian points that are separated 60° from the secondary body.

A really major problem is that Lagrange's solution for the three body problem only works if the tertiary body has a negligible mass. Negligible means small enough that it doesn't perturb the orbit of the secondary body around the primary and that the primary-secondary system can be solved as a simple two body problem. In practice this confines the tertiary into asteroid size masses. If the secondary and tertiary are of comparable masses the L4/5 orbits are wildly unstable. You can try this yourself in any simple orbit simulator.

Another argument against such systems in strictly realistic settings is that the formation of a planet will dominate the mass accretion along its orbit, clearing it from dust and smaller bodies and preventing the formation of another large body.

[gach]

OK, I refreshed my memory about the restricted three-body problem a bit. It's actually a pretty interesting subject.

I wonder how much stability would vary with mass of the primary?

It is dependent from the mass ratio between the primary and secondary. The basic condition is that the secondary has to be roughly 4% of the mass of the primary at most for there to exist stable orbits around the L4 and L5 points. This is a very liberal condition and includes all Sun-planet and planet-satellite pairs in the solar system. The only more commonly known pair in the solar system failing to satisfy this condition is the Pluto-Charon pair which thus lacks stable triangular Lagrange points.

You can see the mathematical treatment of the stability criterion here. Other effects such as additional massive bodies in the system will slightly affect the stability of L4 and L5 but these effects are small.

I'm trying to find free full-text of some of his supporting documentation. $35.00 an article is... a bit steep.

What papers did you have in mind? I could access most of the old papers from home without a problem, though it might be that the journal sites remember that I usually access them from our university network.

I can give two other relevant papers concerning orbits around the triangular Lagrange points. Weissman & Wetherill (1974) have two plots (Figs. 1 & 2) that show the general shapes of bound orbits around L4 and L5 for the Sun-Earth mass ratio. You can see how large the amplitude or the orbits can be, even up to crossing L3 over to the opposite triangular point thus forming a horseshoe orbit.

A fairly famous example of horseshoe orbits are the Saturnian moons Epimetheus and Janus. These orbit Saturn with with orbits differing only 50 km in radius. Despite both of the moons being larger than this they never collide due to being on horseshoe orbits with respect to each other.

Zhang & Innanen (1988) show many more numerical integrations of the tadpole and horseshoe like orbits as well as some other related orbit types. Their orbit calculations also show the radial oscillations that the orbits of objects around L4 and L5 often show.