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Tom Mazanec
2004-May-14, 02:55 AM
Is it possible for two planets to co-orbit a star as Janus and Epimetheus do Saturn? What would be the time period of their switching places, if they could (Janus and Epimetheus are 4 years)?

kenneth rodman
2004-May-14, 06:58 AM
well you just answered your own question. If 2 moons co-orbit a planet, why couldnt 2 planets co-orbit a star. same thing, bigger scale. as t time period, depends on size of star, the planets orbital positions, ect

eburacum45
2004-May-14, 12:33 PM
I disagree; there are no naturally occuring trojan planets in our system, despite hundreds of trojan asteroids; so something suggests that this arrangement would be very rare.

Moose
2004-May-14, 12:54 PM
I disagree; there are no naturally occuring trojan planets in our system, despite hundreds of trojan asteroids; so something suggests that this arrangement would be very rare.

Not really. You have to consider the number of asteroids whose orbits we understand versus the number of planets whose orbits we know.

Right now, we know that nine planets don't have this characteristic. That isn't really enough to express any kind of empirical expectation on what planets can or cannot do.

Any kind of answer to the negative would almost have to be mathematical and would have to demonstrate why asteroids and moons can interact gravitationally to regularly swap orbits, but planets cannot.

Kullat Nunu
2004-May-14, 02:34 PM
If I recall correctly, a trojan must be much less massive than the object whose Lagrangian point it occupies.

So planets in Lagrangian points might not exist, but there have been searches for trojan planets in binary star systems (planet being in secondary star's Lagrangian point).

Cougar
2004-May-14, 11:24 PM
Is it possible for two planets to co-orbit a star as Janus and Epimetheus do Saturn? What would be the time period of their switching places, if they could (Janus and Epimetheus are 4 years)?
Wow, I had a computer model do this once, and I thought I'd stumbled onto something really extraordinary, but here two of Saturn's moons are doing this naturally. :roll: It makes sense they would both be about the same size, with one in a slightly lower and one in a slightly higher orbit - in this case one is 151,422 km and the other is 151,472 km from Saturn. Then when they happen to come close to each other in their orbits, one is gravitationally pulled down (temporarily) and the other is pulled up, swapping orbits! This sort of dynamic might actually be fairly stable with the right initial conditions... and a minimum of perturbations from the outside.

Celestial Mechanic
2004-May-15, 05:03 AM
If I recall correctly, a trojan must be much less massive than the object whose Lagrangian point it occupies.
I'm not sure if this is true. I think both Trojan planets could be of comparable mass, just small in comparison to the star.

So planets in Lagrangian points might not exist, but there have been searches for trojan planets in binary star systems (planet being in secondary star's Lagrangian point).
How would such a planet be searched for? It would have the same period as the secondary. Remember that our current search techniques look for small, periodic changes to the radial velocity of the star. Maybe we could see this as a difference in the phases of the radial velocities of the two stars, that is, the maximum recession velocity of the secondary star away from us does not coincide with the maximum velocity of the primary star towards us. I'm not sure that our measurements are precise enough to pick this up.

There is one condition on the masses that impacts the existence of Trojan orbits. The secondary body must have a mass less than 0.0385 that of the primary. That pretty much rules out advanced life on such a planet, since if the secondary star has a mass near the minimum for a star (about 0.1 solar masses), the primary would be more than 27 times as massive, that is, more than 2.7 solar masses, and such stars don't live very long.

Kullat Nunu
2004-May-15, 07:25 AM
If I recall correctly, a trojan must be much less massive than the object whose Lagrangian point it occupies.
I'm not sure if this is true. I think both Trojan planets could be of comparable mass, just small in comparison to the star.

Okay, then I remembered it wrong. #-o


How would such a planet be searched for? It would have the same period as the secondary. Remember that our current search techniques look for small, periodic changes to the radial velocity of the star. Maybe we could see this as a difference in the phases of the radial velocities of the two stars, that is, the maximum recession velocity of the secondary star away from us does not coincide with the maximum velocity of the primary star towards us. I'm not sure that our measurements are precise enough to pick this up.

No, radial velocity doesn't help us here. But if we see the stars eclipsing each other, also the trojan planet will transit the primary star.

See for example this link (http://www.trojanplanets.appstate.edu/).

Short-period eclipsing stars are good candidates for having transiting planets that orbit both the stars, as it is most likely that the planet orbits on the same plane as the stars.


There is one condition on the masses that impacts the existence of Trojan orbits. The secondary body must have a mass less than 0.0385 that of the primary.

Thanks for making this clear. :)

Eroica
2004-May-15, 04:32 PM
If I recall correctly, a trojan must be much less massive than the object whose Lagrangian point it occupies.
I'm not sure if this is true. I think both Trojan planets could be of comparable mass, just small in comparison to the star.

Mad Scientist (http://www.badastronomy.com/mad/2000/lagrange.html)

Another caveat is that the [Lagrangian]equations are only stable if the two objects are much more massive than the third. If you put a massive object in the L4 or L5 points, the gravity of the object affects the other two, throwing things off. So as long as the third object is low-mass compared to the other two, things are stable.
If I'm reading this right, it seems to bear out what Kullat Nunu is saying.

milli360
2004-May-15, 04:55 PM
If I'm reading this right, it seems to bear out what Kullat Nunu is saying.
From what I've seen of the problem, I would tend to agree, but I haven't gone into it very deeply. What about it, CM?

Kullat Nunu
2004-May-15, 08:09 PM
From the link (http://www.trojanplanets.appstate.edu/Intro.htm) I provided:


The condition for the L4 and L5 points to be very stable, not letting the third body wander too far, requires that the more massive of the two stars be more than twenty times the mass of the less massive star. While individual stars are found in a range in mass this great, no known binary systems contain such a pair. However, the system is still stable if it does not meet this criterion, as long as there are no additional massive objects to perturb the planet.

So they agree with Celestial Mechanic.

Looks like the secondary must be much less massive than the primary and the trojan object much less massive than the secondary.

Celestial Mechanic
2004-May-16, 03:53 AM
I wonder if any research has been done on the stability of L4 and L5 with a non-zero mass for the third body. How much mass could we consider negligible? For example, the mass of Jupiter is about 1/1047 that of the Sun, well below the limit of 0.0385 for stability. Could we put an Earth-size mass (1/330,000 of the Sun's mass) at that point? Mars-size (1/3,000,000 of the Sun's mass)? Pluto-size (1/140,000,000)?

I don't know if any research exists on this question.

eburacum45
2004-May-26, 10:25 AM
Sorry to dig up this weeks old thread, but I've just found this PDF about the impact that formed the Moon billions of years ago.

http://cul.arxiv.org/PS_cache/astro-ph/pdf/0405/0405372.pdf

the gist of it is the Mars sized impactor, which some people have called Orpheus (but not in this link) actually formed in Earth's L4 or L5 point, then was perturbed into a slightly larger or smaller orbit which eventuallycaused it to impact the Earth.

Celestial Mechanic
2004-May-26, 05:30 PM
Thank you for posting the link, eburacum45. I found that paper last week, but I just haven't had time to write up this answer until now. In that article they gave a criterion for stability of L4/L5 when all three objects have non-zero masses.

If m1, m2, and m3 are the masses of the three objects, the motion is stable if 27*(m1*m2+m2*m3+m3*m1) < (m1+m2+m3)^2.

If you set m1 equal to mu, m2 = 1-mu, and m3 equal to zero you get the stability criterion for the restricted problem as it is usually stated, which is 27*mu*(1-mu) < 1.

One interesting consequence is we could move Saturn to Jupiter's L4 point (don't ask me how! :o ) and the motion would be stable.

sol_g2v
2004-May-27, 02:22 AM
Sorry to dig up this weeks old thread, but I've just found this PDF about the impact that formed the Moon billions of years ago.

http://cul.arxiv.org/PS_cache/astro-ph/pdf/0405/0405372.pdf

the gist of it is the Mars sized impactor, which some people have called Orpheus (but not in this link) actually formed in Earth's L4 or L5 point, then was perturbed into a slightly larger or smaller orbit which eventuallycaused it to impact the Earth.

I like this understated paragraph from the conclusion on page 57: "A mythical observer on the earth would observe the growing Mars-sized
impactor as a morning or evening star 60o away from the Sun. Then it would
start to move."

Cue ominous music. :o